MATHEMATICS

CLASS-XII

MATRICES

CONTENTS
KEY CONCEPTS — Page-2-7

PROFICIENCY TEST — Page-8-11

EXERCISE-I — Page-12-13

EXERCISE-II — Page-14-16

EXERCISE-III — Page-17-18

EXERCISE-IV — Page-19-21

EXERCISE-V — Page-22-26

ANSWER KEY — Page-27-28

 KEY CONCEPTS
MATRICES
USEFUL IN STUDY OF SCIENCE, ECONOMICS AND ENGINEERING

1. Definition : Rectangular array of m n numbers . Unlike determinants it has no value.

 a 11 a 12 ...... a 1n   a 11 a 12 ...... a 1 n 
   
 a 21 a 22 ...... a 2 n   a 21 a 22 ...... a 2 n 
A =  : or  :
: : :  : : : 
   
a m1 a m2 ...... a m n   a m1 a m 2 ...... a mn 

Abbreviated as : A = a i j   1  i  m ; 1  j  n, i denotes the row and j denotes the column is

called a matrix of order m × n.

2. Special Type Of Matrices :

(a) Row Matrix : A = [ a11 , a12 , ...... a1n ] having one row . (1 × n) matrix.
(or row vectors)
 a 11 
 
 a 21 
(b) Column Matrix : A =  having one column. (m × 1) matrix
: 
(or column vectors)  
 a m1 
(c) Zero or Null Matrix : (A = Om  n)
An m  n matrix all whose entries are zero .

 0 0  0 0 0
   
A =  0 0 is a 3  2 null matrix & B =  0 0 0 is 3  3 null matrix
 0 0   0 0 0
(d) Horizontal Matrix : A matrix of order m × n is a horizontal matrix if n > m.

1 2 3 4
2 5 1 1
  2 5
1 1
(e) Verical Matrix : A matrix of order m × n is a vertical matrix if m > n. 
3 6
 
2 4
(f) Square Matrix : (Order n)

If number of row = number of column  a square matrix.

Note (i) In a square matrix the pair of elements aij & aj i are called Conjugate Elements .
e.g.
 a 11 a 12 
 
 a 21 a 22 
(ii) The elements a11 , a22 , a33 , ...... ann are called Diagonal Elements . The line along which
the diagonal elements lie is called " Principal or Leading " diagonal.
The qty  ai i = trace of the matrice written as , i.e. tr A

[2]
 Square Matrix
Triangular Matrix Diagonal Matrix denote as
ddia (d1 , d2 , ....., dn) all elements
except the leading diagonal are zero
 1 3  2  1 0 0
   
A = 0 2 4 ; B =  2  3 0 diagonal Matrix Unit or Identity Matrix
   
0 0 5  4 3 3
Upper Triangular Lower Triangular  d1 0 0 
0  1 if i  j
ai j = 0  i > j ai j = 0  i < j
 d2 0  aij =  
 0 if i  j
Note that : Minimum number of zeros in  0 0 d 3 
a triangular matrix of Note: (1) If d1 = d2 = d3 = a Scalar Matrix
order n = n(n–1)/2 (2) If d1 = d2 = d3 = 1 Unit Matrix
Note: Min. number of zeros in a diagonal matrix of order n = n(n – 1)
"It is to be noted that with square matrix there is a corresponding determinant formed by the elements of A in the
same order."

3. Equality Of Matrices :
Let A = [a i j ] & B = [b i j ] are equal if ,
(i) both have the same order . (ii) ai j = b i j for each pair of i & j.
4. Algebra Of Matrices :
Addition : A + B = a ij  bi j  where A & B are of the same type. (same order)
(a) Addition of matrices is commutative.
i.e. A + B = B + A A=mn ; B=mn
(b) Matrix addition is associative .
(A + B) + C = A + (B + C) Note : A , B & C are of the same type.
(c) Additive inverse.
If A + B = O = B + A A = mn
5. Multiplication Of A Matrix By A Scalar :

a b c  ka k b kc 
If A = b c a ; k A =  kb kc ka 
 c a b  kc ka kb 
  
6. Multiplication Of Matrices : (Row by Column)
AB exists if , A = m  n & B= np
23 33
AB exists , but BA does not  AB  BA
 A  prefactor
Note : In the product AB , 
 B  post factor

 b1 
b 
2
A = (a1 , a2 , ...... an) & B =  : 
 
 b n 
1n n1
A B = [a1 b1 + a2 b2 + ...... + an bn]
n
 
If A = a i j m  n & B = bi j   n  p matrix , then (A B)i j = 
r 1
ai r . br j

[3]
Properties Of Matrix Multiplication :
1. Matrix multiplication is not commutative .

1 1 1 0 1 0 1 1
A =
 0 0  ; B =  0 0  ; AB =  0 0 
   
; BA =  0
 0 
 AB  BA (in general)
1 1  1 1   0 0
2. AB =  2 2   1 1 =  0 0   AB = O 
 A = O or B = O

Note: If A and B are two non- zero matrices such that AB = O then A and B are called the divisors of zero.
Also if [AB] = O  | AB |  | A | | B | = 0  | A | = 0 or | B | = 0 but not the converse.
If A and B are two matrices such that
(i) AB = BA  A and B commute each other
(ii) AB = – BA  A and B anti commute each other
3. Matrix Multiplication Is Associative :
If A , B & C are conformable for the product AB & BC, then
(A . B) . C = A . (B . C)
4. Distributivity :
A (B  C)  A B  A C 
Provided A, B & C are conformable for respective products
s
(A  B) C  A C  BC
5. POSITIVE INTEGRAL POWERS OF A SQUARE MATRIX :
For a square matrix A , A2 A = (A A) A = A (A A) = A3 .
Note that for a unit matrix I of any order , Im = I for all m  N.

6. MATRIX POLYNOMIAL :
If f (x) = a0xn + a1xn – 1 + a2xn – 2 + ......... + anx0 then we define a matrix polynomial
f (A) = a0An + a1An–1 + a2An–2 + ..... + anIn
where A is the given square matrix. If f (A) is the null matrix then A is called the zero or root of the polynomial
f (x).

DEFINITIONS :
(a) Idempotent Matrix : A square matrix is idempotent provided A2 = A.
Note that An = A  n > 2 , n  N.
(b) Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order m, m  N, if
Am = O , Am–1  O.

(c) Periodic Matrix : A square matrix is which satisfies the relation AK+1 = A, for some positive integer K, is a
periodic matrix. The period of the matrix is the least value of K for which this holds true.
Note that period of an idempotent matrix is 1.

(d) Involutary Matrix : If A2 = I , the matrix is said to be an involutary matrix.

–1
Note that A = A for an involutary matrix.

7. The Transpose Of A Matrix : (Changing rows & columns)

Let A be any matrix . Then , A = ai j of order m  n
 A or A = [ aj i ] for 1  i  n & 1  j  m of order
T n  m

Properties of Transpose : If AT & BT denote the transpose of A and B ,

(a) (A ± B) = A ± BT
T T ; note that A & B have the same order.
IMP. (b) (A B)T = BT AT A & B are conformable for matrix product AB.
(c) (AT )T = A
(d) (k A)T = k AT k is a scalar .
T T T
General : (A1 , A2 , ...... An)T = A n , ....... , A 2 , A1 (reversal law for transpose)

[4]
8. Symmetric & Skew Symmetric Matrix :
A square matrix A = a 
ij is said to be ,
symmetric if ,
ai j = aj i  i & j (conjugate elements are equal) (Note A = AT)
n ( n  1)
Note: Max. number of distinct entries in a symmetric matrix of order n is .
2
and skew symmetric if ,
ai j =  aj i  i & j (the pair of conjugate elements are additive inverse
of each other) (Note A = –AT )
Hence If A is skew symmetric, then
ai i =  ai i ai i = 0  i
Thus the digaonal elements of a skew symmetric matrix are all zero , but not the converse .

Properties Of Symmetric & Skew Matrix :

P  1 A is symmetric if AT = A
A is skew symmetric if AT =  A
P  2 A + AT is a symmetric matrix
A  AT is a skew symmetric matrix .
Consider (A + AT )T = AT + (AT )T = AT + A = A + AT
T
A + A is symmetric .
Similarly we can prove that A  AT is skew symmetric .

P3 The sum of two symmetric matrix is a symmetric matrix and

the sum of two skew symmetric matrix is a skew symmetric matrix .
Let AT = A ; BT = B where A & B have the same order .
T
(A + B) = A + B
Similarly we can prove the other

P4 If A & B are symmetric matrices then ,

(a) A B + B A is a symmetric matrix
(b) AB  BA is a skew symmetric matrix .

P5 Every square matrix can be uniquely expressed as a sum of a symmetric and a skew symmetric matrix.
1 1
A = (A + AT ) + (A  AT )
2 2

P Q
Symmetric Skew Symmetric

9. Adjoint Of A Square Matrix :

 a11 a12 a13 

 
Let A=  
aij =  a 21 a 22 a 23  be a square matrix and let the matrix formed by the cofactors
a a 33 
 31 a 32
 C11 C12 C13 
 
of [ai j ] in determinant A is =  C 21 C 22 C 23  .
C C33 
 31 C32
 C11 C 21 C 31 
 
Then (adj A) =  C C 22 C32 
12
C 
 13 C 23 C33 

[5]
V. Imp. Theorem : A (adj. A) = (adj. A).A = |A| In , If A be a square matrix of order n.
Note : If A and B are non singular square matrices of same order, then
(i) | adj A | = | A |n – 1
(ii) adj (AB) = (adj B) (adj A)
(iii) adj(KA) = Kn–1 (adj A), K is a scalar
Inverse Of A Matrix (Reciprocal Matrix) :
A square matrix A said to be invertible (non singular) if there exists a matrix B such that,
AB = I = BA
B is called the inverse (reciprocal) of A and is denoted by A 1 . Thus
A 1 = B  A B = I = B A .
We have , A . (adj A) = A In
A 1 A (adj A) = A 1 In 

In (adj A) = A 1 A In
(adj A)
 A 1 =
|A|
Note : The necessary and sufficient condition for a square matrix A to be invertible is that A 0.
Imp. Theorem : If A & B are invertible matrices ofthe same order , then (AB) 1 = B 1 A 1. This is reversal law for
inverse.
Note :
(i) If A be an invertible matrix , then AT is also invertible & (AT ) 1 = (A 1)T .

(ii) If A is invertible, (a) (A 1) 1 = A ; (b) (Ak) 1 = (A 1)k = A–k, k  N

(iii) If A is an Orthogonal Matrix. AAT = I = AT A

(iv) A square matrix is said to be orthogonal if , A 1 = AT .

1
(v) | A–1 | =
|A|
SYSTEM OF EQUATION & CRITERIAN FOR CONSISTENCY
GAUSS - JORDAN METHOD
x+y+z = 6
xy+z=2
2x + y  z = 1

 x  y z  6
 x  y z   
or   =  2
 2x  y z  1

1 1 1  x 6
 1 1 1   y  2
    = 1
 2 1 1  z  

AX = B  A 1 A X = A 1 B

(adj. A).B
X = A 1 B = .
|A|

[6]
Note :
(1) If A 0, system is consistent having unique solution
(2) If A 0 & (adj A) . B  O (Null matrix) ,
system is consistent having unique non  trivial solution .
(3) If A 0 & (adj A) . B = O (Null matrix) ,
system is consistent having trivial solution .
(4) If A  = 0 , matrix method fails

If (adj A) . B = null matrix = O If (adj A) . B  O

Consistent (Infinite solutions) Inconsistent (no solution)

[7]
 PROFICIENCY TEST-01
1. In the following, upper triangular matrix is

 1 0 0 5 4 2  2 1
    0 2 3   
(A) 0 2 0 (B) 0 0 3 (C)   (D) 0 3
    0 0 4  
3 0 3 0 0 1 0 0

5 2  2 3 
2. If A    and B    , then |2A – 3B| equals
1 0 5 – 1

(A) 77 (B) –53 (C) 53 (D) –77

3. For a square matrix A = [aij], aij = 0, when i  j, then A is

(A) unit matrix (B) scalar matrix (C) diagonal matrix (D) None of these

4. If A and B are matrices of order m × n and n × n respectively, then which of the following are defined
(A) AB, BA (B) AB, A2 (C) A2, B2 (D) AB, B2

– 1 5 
 – 1 0 2  
5. If A    and B   2 7  , then
 3 1 2  
 3 10 

(A) AB and BA both exist (B) AB exists but not BA

(C) BA exists but not AB (D) Both AB and BA do not exist

6. If A is a matrix of order 3 × 4, then both ABT and BT A are defined if order of B is

(A) 3 × 3 (B) 4 × 4 (C) 4 × 3 (D) 3 × 4

 0 5 – 7
 
7. Matrix  – 5 0 11  is a
 
 7 – 11 0 

(A) Diagonal matrix (B) Upper triangular matrix

(C) Skew-symmetric matrix (D) Symmetric matrix

8. If A is symmetric as well as skew symmetric matrix, then

(A) A is a diagonal matrix (B) A is a null matrix
(C) A is a unit matrix (D) A is a triangular matrix

9. If A is symmetric matrix and B is a skew-symmetric matrix, then for n  N, false statement is

(A) An is symmetric when n is odd (B) An is symmetric only when n is even
(C) Bn is skew symmetric when n is odd (D) Bn is symmetric when n is even

10. Let A be a square matrix. Then which of the following is not a symmetric matrix
(A) A + AT (B) AAT (C) ATA (D) A – AT

[8]
 1 1
11. If A    and n  N, then An is equal to
1 1

(A) 2nA (B) 2n – 1 A (B) nA (D) None of these

12. If A = [aij] is scalar matrix of order n × n such that aii = k for all i, then |A| equals
(A) nk (B) n + k (C) nk (D) kn

3 – 4
13. If A    , then for every positive integer n, An is equal to
1 – 1

1  2n 4n  8  1  2n – 4n  1– 2n 4n 
(A)  (B)   (C)  (D) None of these
 n 1  2n   n 1 – 2n  n n  2

14. If A is any skew-symmetric matrix of odd orders, then |A| equals

(A) –1 (B) 0 (C) 1 (D) None of these

 0 c – b a 2 ab ac 
   
15. If A   – c 0 a  and B  ab b 2 bc  then AB is equal to
   
 b – a 0  ac bc c 2 
 

(A) A (B) B (C) an Identity matrix (D) a Null matrix

16. If A and B are two square matrices of the same order and (A + B)n = nC0 An + nC1 An–1 B + nC2 An–2 B2 + ......
+ nCn Bn (n is a positive integer), then :
(A) AB = – BA (B) AB = BA (C) |A| = 0 or |B| = 0 (D) holds always

cos  sin    1 0
17. If A =   ,B=   , C = ABAT , then AT CkA is equal to (k  N, k > 5)
 sin   cos   1 1

 k 1  k 0  0 1  1 0
(A)   (B)   (C)   (D)  
 1 0  1 1  k 1  k 1

18. If A is a skew symmetric matrix, then B (I – A)(I + A)–1 is :

(A) Orthogonal matrix (B) Symmetric matrix
(C) Involutary matrix (D) Idempotent matrix

19. Let A be a non-singular square matrix satisfying A3 – 8 A = 0 then (A2 + I)–1 equals to :

1 1 2 1 1 2
(A) (9I – A2) (B) (A – 9I) (C) (9I + A2) (D) (A – I)
9 9 9 9

20. Let A be an idempotent matrix and (I + A)n = I + A (n N) then equals to

(A) 2n – 1 (B) 2n (C) 2n–1 (D) 2n + 1

21. Let M and N be two 3 × 3 non-singular symmetric matrices such that MN = NM. If PT denotes the transpose
of a matrix P, then M2N2 (MT N)–1 (MN–1)T is equal to :
(A) –M2 (B) M2 (C) M2N2 (D) MN
[9]
 PROFICIENCY TEST-02
0 1 1  x 
  
1. The root of the equation [x 1 2]  1 0 1 – 1  0 is
  
 1 1 0  1 
1 1
(A) (B) – (C) 0 (D) 1
3 3

2. For square matrices A and B, AB = O, then {O : null matrix}

(A) A = O or B = O (B) A = O and B = O
(C) It is not necessary that A = O and/or B = O (D) None of these

3. If A and B are matrices of order m × n and n × m respectively, then the order of matrix BT (AT )T is
(A) m × n (B) m × m (C) n × n (D) Not defined

1 2 3
 
4. If A  2 3 4 , then the value of adj (adj A) is
 
0 0 2

(A) 4A2 (B) –2A (C) 2A (D) A2

 cos x sin x   1 0
5. If A    and A.(adj A) = k   , then k equals
 – sin x cos x  0 1

(A) sinx cosx (B) 1 (C) sin 2x (D) –1

 1 –2 3 
 
6. If A   4 0 – 1 , then (adj A)23 =
 
 – 3 1 5 

{i.e., the element of (adj A) which belongs to second row and third column}
(A) 13 (B) –13 (C) 5 (D) –5

7. (adj AT ) – (adj A)T equals

(A) |A| I (B) 2|A| I (C) Null matrix (D) Unit matrix

2 3 4 6 1 0
8. If A   , B   , C    , then which of these matrices are invertible ?
 1 3 2 3 0 1

(A) A and B (B) B and C (C) A and C (D) All

9. Which of the following matrices is inverse of itself

1 1 1  1 0 1 0 1 0 
     
1 1 1 3 2  0 0 0   1 1 1
(A)   (B)   (C)   (D)  
1 1 1  4 3   1 0 1 0 1 0

[10]
10. If D is a diagonal matrix with diagonal elements as {d1, d2, d3 ..., dn} in order, then we may represent it as
D = diag (d1, d2, ......., dn). Then Dn equals
(A) D (B) diag (d1n – 1, d2n – 1, ......, dnn – 1)
n n n
(C) diag (d1 , d2 , ......, dn ) (D) None of these

cos  – sin  0
 
11. If A   sin  cos  0 , then
 
 0 0 1

(A) adj A = A (B) adj A = A–1 (C) A–1 = –A (D) None of these

12. If A is invertible matrix, then det (A–1) equals {where, det (B) means determinant of matrix B}
1
(A) det (A) (B) det (A) (C) 1 (D) None of these

13. If A and B are non-zero square matrices of the same order such that AB = O, then {O : null matrix}
(A) Either adj A = O or adj B = O (B) adj A = O and adj B = O
(C) Either |A| = 0 or |B| = 0 (D) |A| = 0 and |B| = 0

14. Let A be an idempotent square matrix, then (I + A)4 is :

(A) I – A (B) I + A (C) I + 15A (D) I

15. If A and B are two square matrices such that B = –A–1BA, then (A + B)2 =
(A) A2 + 2BA + B2 (B) A2 + B2 (C) A2 + 2AB + B2 (D) A2 – B2

16. If A is a diagonal matrix of order 3 × 3 is commutative with every square matrix of order 3 × 3 and trace (A) =
12 then |A| equals to
(A) 12 (B) 16 (C) 32 (D) 64

17. If A and B are two matrices of 3 × 3 such that AB = 0 and A2 + B = I then |A2 + B2| equals to :
(A) 1 (B) –1 (C) 0 (D) 3

 1  1 0
 
 1 1 1
18. If A =  , then A2 + A–1 =
 0 1 1

(A) I – 3A (B) 3A – I (C) 3A + I (D) I + A

 1 2 3
 
2 3 4
19. If A =  , then |adj (adj2A)| =
5 6 8

(A) 212 (B) 216 (C) 28 (D) 232

 x 3 2
 
1 y 4
20. If A =  , x y z = 50 and 8x + 4y + 3z = 30, then A(adj A) is equal to :
2 2 z 

(A) 64 I (B) I (C) 48 I (D) 16 I

[11]
 EXERCISE-I

1 2 2 
 
2 1  2
1. If 5A =  and AAT = I then find the value of y – x :
 x 2 y 

2. Find the value of x and y that satisfy the equations.

 3  2 3 3
 3 0   y y  = 3y 3y
 2 4   x x  10 10 

a b p 0
3. Let A =  c
 d  and B = q   0 . Such that AB = B and a + d = 5050. Find the value
of (ad – bc).

0 1 8 6 4 2
0
4. Define A = 3 0 . Find a vertical vector V such that (A + A + A + A + I)V = 11

(where I is the 2 × 2 identity matrix).
1 0 2
5. If, A = 0 2 1 , then show that the maxtrix A is a root of the polynomial f (x) = x3 – 6x2 + 7x + 2.
 2 0 3

1 2 a b
6. If the matrices A = 3 4 and B =  c d 

db
(a, b, c, d not all simultaneously zero) commute, find the value of . Also show that the
acb
   2 3
matrix which commutes with A is of the form    

a b 
7. If  c 1  a  is an idempotent matrix. Find the value of f(a), where f(x) = x– x 2, when bc = 1/4. Hence
 
otherwise evaluate a.
1 1
8. If the matrix A is inv olutary, show that (I + A) and (I – A) are idempotent and
2 2
1 1
(I + A)· (I – A)=O.
2 2
1 0
9. Show that the matrix A = 2 1 can be decomposed as a sum of a unit and a nilpotent marix. Hence

2007
evaluate the matrix
1 0 .
 2 1

10. Find number of 2 × 2 matrices with real number as its elements satisfying A + AT = I and AAT = I.

0 1  1
11. Let X be the solution set of the equation I, where A =  4  3 4  and I is the corresponding unit
Ax =
 3  3 4 
matrix and x  N then find the minimum value of  (cos x   sin x ) ,   R.

[12]
  3 a  1  d 3 a 
12. A =2 5 c  is Symmetric and B =  b  a e  2b  c  is Skew Symmetric, then find AB.
b 8 2   2 6  f 
  
Is AB a symmetric, Skew Symmetric or neither of them. Justify your answer.

13. A is a square matrix of order n.

l = maximum number of distinct entries if A is a triangular matrix
m = maximum number of distinct entries if A is a diagonal matrix
p = minimum number of zeroes if A is a triangular matrix
If l + 5 = p + 2m, find the order of the matrix.

14. If A is an idempotent non zero matrix and I is an identity matrix of the same order, find the value of n, n
 N, such that ( A + I )n = I + 127 A.

 1 2 5
15. Consider the two matrices A and B where A = 4 3 ; B =   3 . If n(A) denotes the number of elementss

in A such that n(XY) = 0, when the two matrices X and Y are not conformable for multiplication. If C =

 n (C) | D |2  n ( D)


(AB)(B'A); D = (B'A)(AB) then, find the value of  .
 n ( A)  n ( B) 

16. Let A and B are square matrices, both of order 3 × 3 satisfying A2 + B4 = (AT )2. Find the value of |B|
(AT is transpose matrix of A).

17. Let A and B two non-singular matrices such that (AB)2 = A2B2 then BA2B–1 = An. Find the value of n.

 3  2
18. Let A =   and P be a 2 × 2 matrix such that PPT = I .
 0 1 

If Q = PAPT and R = PT Q8 P = [ri j]2×2 . find r11

19. If A and B are two square matrices of order 3 such that AB = BA and A2 = AB + 2B2. If the matrices A + B and
B are non-signular, final |AB–1|.

 3 0 2  2
   
1 x 5  b  and C = [3
20. Let A =  ;B= 5 1]. Find the number of integral values of 'b' for which
 2 0 x 2    1

tr(ABC)  –18 x R.

[13]
 EXERCISE-II
1. A3 × 3 is a matrix such that | A | = a, B = (adj A) such that | B | = b. Find the value of (ab2 + a2b + 1)S

1 a a2 a3
where S =  3  5  ...... up to , and a = 3.
2 b b b

 4 4 5 
2. For the matrix A =   2 3  3 find A–2.
 3  3 4 

1 1 1
 2 3 1 0 1
3. Given A =  2 4 1 , B = 3 4 . Find P such that BPA = 0 1 0
 2 3 1    

 1 3 5
4. Giv en the matrix A =  1  3  5 and X be the solution set of the equation A x = A,
  1 3 5 

 x3 1 
where x  N – {1}. Evaluate   3 
 x  1  where the continued product extends  x  X.
 

cos x  sin x 0
5. If F(x) =  sin x cos x 0 then show that F(x). F(y) = F(x + y)
 0 0 1 

Hence prove that [ F(x) ]–1 = F(– x).

6. Use matrix to solve the following system of equations.

x  y z3 x  y z3 x  y z3

(i) x  2 y 3z4 (ii) x  2 y3z4 (iii) x  2 y3z4
x  4 y 9 z  6 2 x 3 y  4 z 7 2x 3y 4z 9

7. Let A be a 3 × 3 matrix such that a11 = a33 = 2 and all the other aij = 1. Let A–1 = xA2 + yA + zI then find the
value of (x + y + z) where I is a unit matrix of order 3.

2 1 3 2   2 4 
8. Find the matrix A satisfying the matrix equation, 3 2 . A . 5 3 =  3 1 .


k m
9. If A =   and kn  lm ; then show that A2 – (k + n)A + (kn – lm) I = O.
l n
Hence find A–1.

[14]
  2 1 9 3
10. Given A=  2 1 ; B= 3 1 . I is a unit matrix of order 2. Find all possible matrix X in the following cases.
   
(i) AX = A (ii) XA = I (iii) XB = O but BX  O.

 1 2
11. If A = 2 4 then, find a non-zero square matrix X of order 2 such that AX = O. Is XA = O.


 1 2
If A = 2 3 , is it possible to find a square matrix X such that AX = O. Give reasons for it.


 3  2 1  x   b 
12.
 
Determine the values of a and b for which the system 5  8 9 y   3 
 2 1 a   z    1

(i) has a unique solution ; (ii) has no solution and (iii) has infinitely many solutions

1 2 3 1  1 2  x1 x2 
13. If A = 3 ;B= 1 0 ; C = 2 4 and X = x x 4  then solve the following matrix equation.
 4  3
(a) AX = B – I (b) (B – I)X = IC (c) CX = A

14. If A is an orthogonal matrix and B = AP where P is a non singular matrix then show that the matrix
PB–1 is also orthogonal.

3  4 a b
15. Consider the matrices A = 1 and B = APT and
0 1 and let P be any orthogonal matrix and Q = PAP
  1 
R = PT QKP also S = PBPT and T = PT SKP
Column I Column II
(A) If we vary K from 1 to n then the first row (P) G.P. with common ratio a
first column elements at R will form
(B) If we vary K from 1 to n then the 2nd row 2nd (Q) A.P. with common difference 2
column elements at R will form
(C) If we vary K from 1 to n then the first row first (R) G.P. with common ratio b
column elements of T will form
(D) If we vary K from 3 to n then the first row 2nd column (S) A.P. with common difference – 2.
elements of T will represent the sum of

i  j ; if i  j
16. Let A = [aij]3×3 be a matrix, where aij =  2 . If Cij be the co-factors of aij in the matrix A. If B =
i ; if i  j

[bij]3×3 be a matrix such that bij = a

k 1
ik C jk . Find the value of |B|.

17. Consider a square matrix A of order 3 such that |A| = 4. If adj(adj A) = A, find the value of .

[15]
18. If A be a non-singular matrix of order 2 such that A + adjA = A–1. Find the value of |3A–1|.

19. If A and B are square matrices of the same order such that |A| = |B| =  ( > 0) and A(adj A + adj B) = B. Find
the value of |A + B|

2 3 
20. If A =   . Find the value of det (A 4) + det (A10 – (adj(2A))10).
0  1

[16]
 EXERCISE-III
  
1. If A =   and A2 = 2 (2 × 2 Identity matrix), then ,  and  will satisfy the relation
  –  

(A) 1 + 2 +  = 0 (B) 1 – 2 +  = 0 (C) 1 + 2 –  = 0 (D) –1 + 2 +  = 0

 cos  sin  
2. If A     , then which of following statement is true
– sin  cos  

 cos n  sinn    cos n sin n 

n
(A) A . A = A and ( A  )    (B) A . A = A and ( A  )n   
  
 – sin  cos  
n n
 – sin n cos n 
 

 cos n  sinn    cos n sin n 

n n
(C) A . A = A and ( A  )    (D) A . A = A
   and ( A  )   
 – sinn  cos n    – sin n cos n 
 

1 2
3. If M    and M2 – M – I = 0, then  equals
2 3

(A) –2 (B) 2 (C) –4 (D) 4

– 1 2 
4. If A be a matrix such that inverse of 7A is the matrix   , then A equals
 4 – 7 

 1 2  1 4 / 7  1 4  1 2 / 7
(A)   (B)   (C)   (D)  
4 1 2 / 7 1/ 7  2 1 4 / 7 1/ 7 

 0 1
5. If A    and (aI + bA)2 = A, (a > 0), then
 – 1 0

1 1
(A) a  b  2 (B) a  b  (C) a  b  3 (D) a  b 
2 3

6. If A and B are square matrices such that AB = B and BA = A, then A2 + B2 is equal to

(A) 2AB (B) 2BA (C) A + B (D) None of these

–1
 1 – tan   1 tan   a – b
7. If       , then
tan  1  – tan  1  b a 

(A) a = sin 2, b = – cos 2 (B) a = cos 2, b = sin 2

(C) a = sin 2, b = cos 2 (D) a = cos 2, b = – sin 2

[17]
 1 2 3 7 
8. Let the matrices A and B be defined as A    and B    , then the value of determinant of matrix
2 3   1 3
(2A7B–1), is :
(A) 2 (B) 1 (C) –1 (D) –2

9. There are two possible values of A in the solution of the matrix equation
1
 2A  1 5   A  5 B  14 D
 4  , where A, B, C, D, E, F are real numbers. The absolute value of the
 A  2A  2 C   E F 

difference of these two solutions, is :

13 11 17 19
(A) (B) (C) (D)
3 3 3 3

10. If A is a square matrix, and B is a singular matrix of same order, then for a positive integer n, (A–1BA)n equals
(A) A–nBnAn (B) AnBnA–n (C) A–1BnA (D) n(A–1BA)

[18]
 EXERCISE-IV

a b    
1. If A    and A 2    , then [AIEEE 2003]
b a   
(A)  = a2 + b2,  = ab (B)  = a2 + b2,  = 2ab
(C)  = a2 + b2,  = a2 – b2 (D)  = 2ab,  = a2 + b2

 0 0 1
 
2. Let A =  0 1 0  . The only correct statement about the matrix A is : [AIEEE 2004]
 1 0 0 

(A) A is a zero matrix (B) A2 = I

(C) A–1 does not exist (D) A = –I, where I is a unit matrix

 1 1 1   4 2 2
 5 0  
3. Let A  2 1 3  and 10B    . If B is the inverse of A, then  is : [AIEEE 2004]
 
 1 1 1   1 2 3 

(A) –2 (B) 5 (C) 2 (D) –1

4. If A2 – A + I = O, then the inverse of A is : [AIEEE 2005]

(A) A + I (B) A (C) A – I (D) I – A

1 0   1 0
5. If A  
1 1  and I  0 1 , then which one of the following holds for all n  1, by the principle of mathematical
   
induction [AIEEE 2005]
(A) An = nA – (n–1)I n n–1
(B) A = 2 A – (n–1)I n
(C) A = nA + (n –1)I n n–1
(D) A = 2 A + (n–1)I
6. If A and B are square matrices of size n × n such that A2 – B2 = (A – B)(A + B), then which of the following will
be always true ? [AIEEE 2006]
(A) A = B (B) AB = BA
(C) Either A or B is a zero matrix (D) Either A or B is an identity matrix

1 2 a 0 
7. Let A    and B  0 b  , a, b  N. Then [AIEEE 2006]
3 4   
(A) there cannot exists any B such that AB = BA.
(B) there exists more than one but finite numbe of B's such that AB = BA.
(C) there exists exactly one B such that AB = BA.
(D) there exists infinitely many B's such that AB = BA.

5 5  
8. Let A  0  5  . If |A2| = 25, then || equals : [AIEEE 2007]
0 0 5 

(A) 52 (B) 1 (C) 1/5 (D) 5

[19]
9. Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of
diagonal entries of A. Assume that A2 = I. [AIEEE 2008]
Statement 1 : If A  I and A  –I, then detA = –1.
Statement 2 : If A  I and A  –I, then tr(A)  0.
(A) Statement 1 is false, statement 2 is true.
(B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
(C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
(D) Statement 1 is true, statement 2 is false.

10. Let A be a 2 × 2 matrix. [AIEEE 2009]

Statement 1 : adj.(adj A) = A
Statement 2 : |adj A| = |A|
(A) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
(B) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
(C) Statement 1 is true, statement 2 is false.
(D) Statement 1 is false, statement 2 is true.

11. The number of 3 × 3 non-singular matrices with four entries as 1 and all other entries as 0 is : [AIEEE 2010]
(A) at least 7 (B) less than 4 (C) 5 (D) 6

12. Let A be a 2 × 2 matrix with non-zero entries and let A2 = I, where I is a 2 × 2 identity matrix. Define Tr(A) =
sum of diagonal elements of A and |A| = determinant of matrix A. [AIEEE 2010]
Statement 1 : Tr(A) = 0
Statement 2 : |A| = 1
(A) Statement 1 is false, statement 2 is true.
(B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
(C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
(D) Statement 1 is true, statement 2 is false.

13. Let A and B two symmetric matrices of order 3. [AIEEE 2011]

Statement 1 : A(BA) and (AB)A are symmetric matrices
Statement 2 : AB is symmetric matrix if matrix multiplication of A with B is commutative.
(A) Statement 1 is false, statement 2 is true.
(B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
(C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
(D) Statement 1 is true, statement 2 is false.

1 0 0  1 0
     
14. Let A   2 1 0  . If u1 and u2 are column matrices such that Au1   0  and Au2   1  , then u1 + u2 is
 3 2 1 0 0
     

equal to : [AIEEE 2012]

 1  1  1 1

       
1 1 1 1
(A)   (B)   (C)   (D)  
0  1 0  1
       

15. Let P and Q be 3 × 3 matrices P  Q. If P3 = Q3 and P2Q = Q2P, then determinant of (P2 + Q2) is equal to :
[AIEEE 2012]
(A) –2 (B) 1 (C) 0 (D) –1

[20]
  1  3
 
16. If P =  1 3 3 is the adjoint of a 3 × 3 matrix A and |A| = 4, then is equal to :
2 4 4

(A) 0 (B) 4 (C) 11 (D) 5 [JEE Main - 2013]

17. If A is an 3 × 3 non-singular matrix such that AA' = A'A and B = A–1 A', then BB' equals
[JEE Main - 2014]
(A) (B–1)' (B) I + B (C) I (D) B–1

1 2 2 
 
18. If A =  2 1 2  is a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the
a 2 b 

ordered pair (a, b) is equal to: [JEE Main - 2015]

(A) (–2, –1) (B) (2, –1) (C) (–2, 1) (D) (2, 1)

 5a  b 
19. If A =   and A adj A = A AT , then 5a + b is equal to : [JEE Main- 2016]
3 2

(A) –1 (B) 5 (C) 4 (D) 13

 2 – 3
20. If A =   , then adj (3A2 + 12A) is equal to : [JEE Main - 2017]
– 4 1

 51 84  72 – 63   72 – 84  51 63 
(A)   (B)   (C)   (D)  
63 72  – 84 51   – 63 51  84 72 

1 2
21. Let A be a matrix such that A•   is a scalar matrix and |3A| = 108. Then A2 equals : [JEE Main - 2018]
0 3 

 4 32   36 0   4 0 36 32

(A)  0 36  (B)  32 4  (C)  32 36  (D)  0 4 
      

22. Suppose A is any 3 × 3 non-singular matrix and (A – 3I) (A – 5I) = O, where I = I3 and O=O3. If A + A–1 =
4I, then + is equal to : [JEE Main - 2018]
(A) 8 (B) 7 (C) 13 (D) 12

1 0 0 
 
23. Let A = 1 1 0  and B = A20. Then the sum of the elements of the first column of B is [JEE Main - 2018]
1 1 1
(A) 211 (B) 210 (C) 231 (D) 251

[21]
 EXERCISE-V
1. Let X and Y be two arbitrary, 3 × 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 × 3, non-zero,
symmetric matrix. Then which of the following matrices is (are) skew symmetric? [JEE Advanced 2015]
(A) Y3Z4 – Z4Y3 (B) X44 + Y44 (C) X4Z3 – Z3X4 (D) X23 + Y23

3  1  2
 
2. Let P = 2 0   , where   R. Suppose Q = [qij] is a matrix such that PQ = kI, where k  R, k  0
3  5 0 
k k2
and I is the identity matrix of order 3. If q23 = – and det(Q) = , then [JEE Advanced 2016]
8 2
(A)  = 0, k = 8 (B) 4– k + 8 = 0
(C) det(Padj(Q)) = 29 (D) det(Q adj(P)) = 213

 1 0 0
3. Let P   4 1 0 and I be the identity matrix of order 3. If Q = [qij] is a matrix such that P50 – Q = I, then
16 4 1

q31  q32
q21 equals [JEE Advanced 2016]
(A) 52 (B) 103 (C) 201 (D) 205

1  2  x  1 
     
4. For a real number , if the system   1   y    – 1 of linear equations, has infinitely many solutions,
 2  1  z   1 
 
then 1 +  + 2 = [JEE Advanced 2017]

5. Which of the following is (are) NOT the square of a 3 × 3 matrix with real entries ? [JEE-Advanced-2017]

1 0 0  1 0 0   1 0 0   1 0 0
0 1 0  0 1 0   0 1 0  0 1 0 
(A)   (B)   (C)   (D)  
0 0 1 0 0 1  0 0 1 0 0 1

 b1 
 
6. Let S be the set of all column matrices b 2  such that b1, b2, b2   and the system of equations (in real
b 3 
variables)
–x + 2y + 5z = b1
2x – 4y + 3z = b2
x – 2y + 2z = b3
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one

 b1 
 
solution for each b 2   S? [JEE Advanced 2018]
b 3 
(A) x + 2y + 3z = b1, 4y + 5z = b2 and x + 2y + 6z = b3
(B) x + y + 3z = b1, 5x + 2y + 6z = b2 and –2x – y – 3z = b3
(C) –x + 2y – 5z = b1, 2x – 4y + 10z = b2 and x – 2y + 5z = b3
(D) x + 2y + 5z = b1, 2x + 3z = b2 and x + 4y – 5z = b3

[22]
7. Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {–1, 0, 1}. Then, the maximum
possible value of the determinant of P is _________. [JEE Advanced 2018]

 sin 4  – 1 – sin2 
8. Let M=  2  = I + M–1
1  cos  cos 4  

where = () and = () are real numbers, and I is the 2 × 2 identity matrix. If
* is the minimum of the set {() : [0, 2)} and
* is the minimum of the set {() : [0, 2)},
then the value of * + * is : [JEE Advanced 2019]

29 37 17 31
(A) – (B) – (C) – (D) –
16 16 16 16

0 1 a   –1 1 –1
9. Let M   1 2 3  and adj M   8 –6 2 
   
3 b 1  –5 3 –1

where a and b are real numbers. Which of the following options is/are correct? [JEE Advanced 2019]

    1
   
(A) (adj M)–1 + adj M–1 =–M (B) If     2 , then – +  = 3
M
   3 

(C) det(adj M2) = 81 (D) a + b = 3

10. Let x  R and let [JEE Advanced 2019]

 1 1 1 2 x x 
   
P = 0 2 2 , Q = 0 4 0  and R = PQP–1.
0 0 3   x x 6 

Then which of the following options is/are correct?

(A) There exists a real number x such that PQ = QP

2 x x 
 
0 4 0
(B) det R = det  + 8, for all x  R
 x x 5 

 1  1
   
a a
(C) For x = 0, if R   = 6   , then a + b = 5
b  b 

   0 
   
(D) For x = 1, there exists a unit vector  ˆi  ˆj  kˆ for which R     0 
   0 

[23]
11. Let
 1 0 0  1 0 0 0 1 0 
     
0 1 0 0 0 1 1 0 0
P1 = I =  , P2 =  , P3 =  ,
0 0 1 0 1 0  0 0 1

0 1 0  0 0 1 0 0 1
     
0 0 1 1 0 0 0 1 0
P4 =  , P5 =  , P6 = 
 1 0 0  0 1 0   1 0 0 

6 2 1 3 
 
and X=  Pk  1 0 2 PkT [JEE Advanced 2019]
k 1 3 2 1

where PkT denotes the transpose of the matrix Pk. Then which of the following options is/are correct?
(A) X – 30I is an invertible matrix (C) X is a symmetric matrix

1 1
   
(C) If X 1   1 , then  = 30 (D) The sum of diagonal entries of X is 18
1 1

12. Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj(adj
M), then which of the following statements is/are ALWAYS TRUE ? [JEE Advanced 2020]
(A) M = I (B) det M = 1 (C) M2 = I (D) (adj M)2 = I

13. The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 × 2 matrix such that
the trace of A is 3 and the trace of A3 is – 18, then the value of the determinant of A is _____
[JEE Advanced 2020]

14. For any 3 × 3 matrix M, let |M| denote the determinant of M. Let

1 2 3  1 0 0 1 3 2 
E  2 3 4  , P  0 0 1 and F 
 
8 18 13 
 
8 13 18  0 1 0  2 4 3 

If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is (are) TRUE ?
[JEE Advanced 2021]

 1 0 0
(A) F = PEP and P  0 1 0 
2
 
0 0 1

(B) |EQ + PFQ–1] = |EQ| + |PFQ–1|

(C) |(EF)3| > |EF|2
(D) Sum of the diagonal entries of P–1 EP + F is equal to the sum of diagonal entries of E + P–1 FP

15. For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F
be two 3 × 3 matrices such that (I – EF) is invertible. If G = (I – EF)–1, then which of the following statements
is (are) TRUE? [JEE Advanced 2021]
(A) |FE| = |I – FE||FGE| (B) (I – FE)(I + FGE) = I
(C) EFG = GEF (D) (I – FE)(I – FGE) = I

[24]
  5 3
 
16. If M =  2 2  , then which of the following matrices is equal to M2022? [JEE Advanced 2022]
 3 1
– 

 2 2

 3034 3033   3034 – 3033   3033 3032   3032 3031 

(A)   (B)   (C)   (D)  
  3033  3032   3033  3032    3032  3031   – 3031 – 3030 

17. Let  be a real number. Consider the matrix

 0 1
 
A=  2 1  2
 3 1  2
 
If A7 – ( – 1)A6 – A5 is a singular matrix, then the value of 9is _______. [JEE Advanced 2022]

18. Let M = (aij), i, j  {1, 2, 3}, be then 3 × 3 matrix such that aij = 1 if j + 1 is divisible by i, otherwise aij = 0. Then
which of the following statements is(are) true? [JEE Advanced 2023]
(A) M is invertible

 a1   a1    a1 
     
(B) There exists a nonzero column matrix  a 2  such that M  a2  =   a2 
a  a   a 
 3  3  3

 0
 
0
(C) The set {X  R3 : MX = 0}  {0}, where 0 =  
 0
 
(D) The matrix (M – 2I) is invertible, where I is the 3 × 3 identify matrix

 a 3 b  
  
19. Let R =  c 2 d  : a, b, c, d  {0, 3, 5, 7,11,13,17,19 }  . Then the number of invertible matices in R is :
 0 5 0  
  
[JEE Advanced 2023]

20. Let  and  be the distinct roots of the equation x2 + x – 1 = 0. Consider the set T = {1, , }. For a 3 × 3
matrix M = {aij}3×3, define Ri = ai1 + ai2 + ai3 and Cj = a1j + a2j + a3j for i = 1, 2, 3 and j = 1, 2, 3.
Match each entry if List -I to the correct entry in List-II. [JEE Advanced 2024]
List-I List-II
(P) The number of matrices M = (aij)3×3 with (1) 1
all entries in T such that Ri = Cj = 0
for all i, j, is
(Q) The number of symmetric matrices (2) 12
M = (aij)3×3 with all entries in T such that
Cj = 0 for all j, is
(R) Let M = (aij)3×3 be a skew symmetric (3) infinite
matrix such that aij  T for i > j. Then the
number of elements in the set

 x   x   a12 
     
 y  : x, y, z  R, M  y    0  is
 z   z    a 
     23 

[25]
(S) Let M = (aij)3×3 be a matrix with all (4) 6
entries in T such that Ri = 0 for all i.
Then the absolute value of the determinant
of M is
(5) 0
The correct options is :
(A) (P)  (4), (Q)  (2), (R)  (5), (S)  (1)
(B) (P)  (2), (Q)  (4), (R)  (1), (S)  (5)
(C) (P)  (2), (Q)  (4), (R)  (3), (S)  (5)
(D) (P)  (1), (Q)  (5), (R)  (3), (S)  (4)

[26]
 ANSWER KEY
PROFICIENCY TEST-01
1. B 2. B 3. C 4. D 5. A 6. D 7. C
8. B 9. B 10. D 11. B 12. D 13. B 14. B
15. D 16. B 17. D 18. A 19. A 20. A 21. B

PROFICIENCY TEST-02
1. A 2. C 3. D 4. B 5. B 6. A 7. C
8. C 9. B 10. C 11. B 12. B 13. D 14. C
15. B 16. D 17. A 18. B 19. A 20. C

EXERCISE-I
0
3  
1. 1.00 2. x= , y = 2 3. 5049 4. V=   6. 1
2 1
 
11
 1 0
7. f (a) = 1/4, a = 1/2 9.  4014 1  10. 2.00

11. 2 12. AB is neither symmetric nor skew symmetric
13. 4 14. n=7 15. 650 16. 0.00
17. 2.00 18. 81.00 19. 8.00 20. 5.00

EXERCISE-II
 17 4  19 
   4 7  7
  10 0 13 
1. 225 2. 3.   4. 3/2
  21  3 25   3 5 5 

6. (i) x = 2, y = 1, z = 0 ; (ii) x = 2 + k, y = 1  2k, z = k where k  R ;

(iii) inconsistent, hence no solution

1  48  25 1  n  m
7. 1 8.   9.  
19 70 42  kn  m   k 

 a b 
10. (i) X=  for a, b  R ; (ii) X does not exist ;
2  2a 1  2b

a  3a
(iii) X=  a, c  R and 3a + c  0; 3b + d  0
 c  3c 

 2c  2d
11. X=   , where c, d  R – {0}, NO
 c d 

12. (i) a  – 3 , b  R ; (ii) a = – 3 and b  1/3 ; (iii) a = –3 , b = 1/3

  3  3 1 2
13. (a) X=  5 2
 , (b) X =   , (c) no solution 15. (A) Q; (B) S; (C) P; (D) P
 2   1  2

16. (62)3 17. 4.00 18. 18.00 19. 2–n 20. 16.00

[27]
 EXERCISE-III
1. D 2. D 3. D 4. D 5. B 6. C 7. B
8. D 9. D 10. C

EXERCISE-IV
1. B 2. B 3. B 4. D 5. A 6. B 7. D
8. C 9. D 10. B 11. A 12. D 13. C 14. D
15. C 16. C 17. C 18. A 19. B 20. D 21. D
22. A 23. C

EXERCISE-V
1. C,D 2. B, C 3. B 4. 1 5. A,C 6. A, D 7. 4

8. A 9. ABD 10. BC 11. BCD 12. BCD 13. 5.00 14. A,B,D

15. A,B,C 16. A 17. 3.00 18. B,C 19. 3780.00 20. C

[28]