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The document provides an overview of matrices, including definitions, types (such as square, diagonal, identity, upper and lower triangular), and operations (addition, subtraction, multiplication, and transposition). It also covers properties of matrix multiplication, symmetric and skew-symmetric matrices, and the concept of invertible matrices. Additionally, it includes practice exercises and assertion-reason type questions related to matrix theory.
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EERE EEE IEEE EE EID EI EI IDI IDI EI IDI ID II II III II II IID IEEE EO ET
Matrices
Fastracl« Revision
> Arectangular array of mn numbers arranged in mrows and
‘columns s called a matrix of order m x”. The numbers are
called the elements or entries of the matrix.
> The element of throw and/thcolumn|s represented by ay.
> Two matrices are said to be equal, if the order of two
matrices Is equal and corresponding elements are also
equal.
> A matrix having the same number of rows and columns is
called a square matrix.
» A square matrix in which every non-diagonal element Is
zero, is called a diagonal matrix
» A square matrix in which every non-diagonal element is 0
and every diagonal element is 1, is called an identity or unit
matrix.
> A square matrix A=[ay] is sald to be upper triangular
matrix, Fay =O >].
> A square matrix A=[aj] Is said to be lower triangular
matrix, Fay =0 ¥ i If Aand B are two matrices of same order, then their sum.
(A+ 8) is a matrix of same order which is obtained by
adding the corresponding elements of A and 8.
> IF A and B are two matrices of same order, then their
difference (A~ 8) Is equal to A+(-8), Le, sum of matrix A
and the matrix(~8).
» The product ABof two matrices Aand Bcan be determined
only when number of columns in matrix A= number of rows,
in column 8
» IF Aand B are square matrices of the same order, say ‘n'
then both the product AB and BA are defined and each Is a
square matrix of order‘.
>In the matrix product A8, the matrix A is called
pre-multiplier (pre-factor) and 8 is called post-multiplier
(post-Factor)
> Properties of Matrix Multiplication
() AB» BA
ly) ABC) =(AB)C
(i) (8+ 0) 248+ AC
(Ww) (As B)C=AC + BC
() Al=A=lA
> The matrix obtained by interchanging rows into columns
‘and columns into rows of matrix Ais called the transpose
matrix of matrix A, denoted by A" or 4.
» IF the order of matrix Ais m xn, then
order of ” =nxm
» Some Results on Transpose of Matrices
(ayaa
(iy (As a =al sa”
(il) (kay? =kar™
(iw) (AB)! = 87a,
> A square matrix of order n xn is sald to be orthogonal, IF
AB =h= AA.
>» Asquare matrix Als said to be symmetric matriy, if Al = A.
» IF Aand Bare symmetric matrices of the same order, then
the product ABIs symmetric, iff B= AB.
> A square matrix A Is sald to be skew-symmetric matrix, IF
Arend,
» All elements of main diagonal of a skew-symmetric matrix
are zero
» Every square matrix can be expressed uniquely as a sum of
a symmetricand a skew-symmetric matrices.
I alsa square maths then Anes where, = As A
(where kis a constant)
Isasymmetricmatrand@Q = 1(A~A")lsaskewsymmetrc
matrix
> A square matrix A of order m/s called invertible, if there
exists a square matrix B of order m such that
AB =BA=ly
Also, Bis called the inverse matrix of A and is denoted by
ie
» The inverse of every invertible matrix is unique.
» IF Aand Bare two invertible square matrices of same order
then
(aay 28-14
x Practice Exercise
¢ Questions
-Q Multiple ch
1. The number of all possible matrices of order 2x 3
with each entry 4 or 2 is: (CBSE 2021 Tarm-1)
als
64
b6
26
2. IFA=[a, Jis a square matrix of order 2 such that
1, whens # j
ay =
then A
0, when/=j?"e"
(CBSE SQp 2023-24, 21 Term-1, CBSE 2023)
shal foal chal eles) ee
EERE EEE IEEE EE EID EI EI IDI IDI EI IDI ID II II III II II IID IEEE EO ET
3. Amatrix A= (a, ]s,; is defined by:
+3; ij
‘The number of elements in Awhich are more than
5, (€8S€ 2021 Term-1)
a b4
c d6
(c8se2023)
a+b a-2b)] [4 -5
a6. if = , then the value of
Se-d 4c+3d| [11 2:
a+b-c+2d is: (case sop 2021 Term-1)
a8 b.10 c4 a8
3a
02 0
O56. A=|, “J andéa=|_, 54 )then the values
of k,a and b respectively are:
(CBSE SQP 2021 Term-1)
b.-6,-4,-9
d.-6.12.18
a.-6 12-18
6-649
a b\fa -b
7. The product i seat to:
[Ls a\|b a (CBSE 2023)
7 ee 0 ] ofesee Al
OQ a?+b? (a+b? o.
fare 0 al? ®
+b? 0 a6
98. wal) s}e-[f tlanta= 0%, then equate:
2ap "ia 4 {€8SE2023)
atl b-1
cl 42
1
nal? 2° ° 5
wa Aly oo of Ol a xf orf] om
15+ x)
=|,” |such that @A-38)C =, then x =
a3 b4 e-6 a6
11 d][x] [6
gu. tf jo 1 4//y|=|3) then the value of
oo allz| |2
Qx+y=z)is: (cose 2023)
al b.2 3 A
4) *
rl 5 -afy eon +370
3 z
al ba c4 ao
1
ove Ae] a a ak 24
equal to:
aa bua
4 and not? d.lor4
13, If for a square matrix A, A? -A+/=0, then A>
equals: (case 2023)
aa bat
clea oA
qu.
b.b=0
d.d=0
016. If a matrix A=[1 2 3], then the matrix AA’
(where A’isthe transpose of A)is: _(casE2023)
loo
o0 3
{ 0.04)
a6. taal . Jona 231, then:
1 70. (COSE Sop 2021 Tern)
alea?+py=9 bl-a?-f
c3-a?py 20 634074 py=0
017, If matrices A and B are of order 3xn and mx5
respectively, then the order of matrix € =5A+38
2 (c88e Sop 2021 Term-1)
a3x5andmen ——b3x5
353 55
918. Matrix A has m rowsand ( + 5)columns, matrix 8
has m rows and (14~n) columns. If both AB and
BA exists, then:
& ABand BA are square matrices
b. AB and BA are of orders 8x8 and 3x13
respectively
© AB=BA
None of the above
Q19. IF Ais3x 4 matrix and B is a matrix such that A’B
and BA’ are both defined, then the order of matrix
Bis: (cose 2023)
aaxd bax
4x4 4x3
oad
Q20. For the matrix X =|1 0 1), (x? —X)is:
110
(CBSE 2021 Term-1)
a2l bal cl 5) ee
3
oz. U=[2 -3 4, V=|2|,x=[0 2 3] and
4
2
Y =|2| then the value of UV + XY
4
azo b(-20) ¢-20 4 (20)
14 109 j
922. HA=| | then A¥" is equal to:
2a b.2A
100A d.299A
1 0 =1 i .
92a. tar eel qlanta-ae-[ 5 _apthen Ais:
(cose 2020)
ab] eh |
ey “ah |
024. if af al and A=AT, where A” is the
y 0
transpose of the matrix A, then:
(c0se2023)
bxey
dx=5y=0
925. Ifa matrix Ais both symmetric and skew-symmetric,
then Ais necessarily a/an:
(NCERTEXEMPLAR; CBSE2021 Term-1)
a. dlagonal matrix b. zero square matrix
square matrix d. identity matrix
26. If A and B are symmetric matrices of the same
order, then:
a. ABs a symmetric matrix
b.A~Bisa skew-symmetric matrix
©. AB + BAls a symmetric matrix
d, AB ~ BA ls a symmetric matrix
27. Hfa square matrix A=(a,],a, =/? ~ j?is of even
order, then:
a. Als a skew-symmetric matrix
b. Als asymmetric matrix
© Both a. and b,
d. Als neither symmetric nor skewesymmetric
3 x=), ,
2a. tales we 2|'8 asymmetric matrix, then
xe
a4 ba
cA a3
29, If A and B are symmetric matrices and AB = BA,
then A“*Bis a:
a. symmetric matrix
b. skew-symmetric matrix
unit matrix
None of the above
030. If A=|a, ]is a skew-symmetric matrix of order n,
then: (cBse sop 2022.23)
b.o 20¥i)
} doy #0. where i =/
Q31. If A is square matrix such that A? =A, then
(1+4)5-74Ais equal to: (cuSE Sop 2021 Term)
aa bisa cla dl
92. If A is a square matrix and A’=A, then
(+A)? -3Ais equal to: (CBSE 2023)
al bA 2a a3
933. IfA? =A, then (I + A)* is equal to:
alta b+ 4A
cl415A d. None of these
34, IfA® =0,then A? + A+) =
al-A b.(|- ay"
cleay d1+A
35. If AB=A and BA=8, then:
a8 bAsl cAt=A dB? =!
036. IfAB=Aand BA=B, then A? + B” is equal to:
aA+B b.A-B
C2448 d. None of these
037. If A and B are square matrices of size nx n such
that A? ~B? =(A~B) (A+B), then which of the
following will be always true?
a. ither A or Bis a zero matrix
b. Either A or B's an identity matrix
CAsB
1, ABBA
88. IF A,B are non-singular square matrices of the
same order, then (AB™)“? = (case sap 2022-23)
A's bate
BA 4.aB
1-1 0 22 4
Q99. fAs|2 3 4/and@=|-4 2 -4], then:
012 2-15
(CBSE SQP 2021 Torm-1)
1
aAte8 b.AT=68 cB'=8
200
Q.40. The inverse of the matrix X=|0 3 O|is:
004
(CBSE 2021 Torm-1)
v2 0 OF {fi99
a2 ova o] blo lo
o 04 oo}
{f2e° v2 0 0
ozo 3 0 d/o 3 0
004 ooo V4 017 4/2 -1/2 1/2
gatas] 2 3/andAt=|-4 3 |,
Bal 5/2 -3/2 1/2
then the values of a and ¢ are:
all bi-l
cl2 du
42. If A and B are square matrices of the same order
and AB =3/, then A~ is equal to:
1
038 bie
1 Tee
3B" 456
43. IFA and B are square matrices of the same order
such that (A+B) (AB) =A? -B?, then (ABA™*)?
is equal to:
a8 bt
Ate aa?
044, If A? -A+1=0, then the inverse of A is:
al-A bh AW] cA a As!
-@ Assertion & Reason type Questions
| directions (Q. Nos. 45-56): In the following questions, each
) question contains Assertion (A) and Reason (R). Each question
has 4 choices (a), (b), (¢) and (d) out of which only one is
) correct. The choices are:
a. Both Assertion (A) and Reason (R) are true and
) Reason (R) is the correct explanation of
Assertion (A)
b, Both Assertion (A) and Reason (R) are true but
Reason (R) is not the correct explanation of
Assertion (A)
c Assertion (A) Is true but Reason (R) Is false
d. Assertion (A) I false and Reason (R) Is true
/ cannot be
742]
expressed as a sum of symmetric and
skew-symmetric matrices.
45. Assertion (A): Matrix 3x 3,ay
Reason (R): Matrix 3x3,a, =——~ is neither
742i
symmetric nor skew-symmetric,
ky is]
0.46. Assertion (A): Scalar matrix Aroule{s, fe}
where, k is a scalar, is an identity matrix when
kat,
Reason (R): Every
matrix.
not a scalar
30
047. Assertion (A):|0 4 0)
007
is a diagonal matrix.
(0.48. Assertion (A):
matrix
Reason (R): If B = [b; ];,.. is a row matrix, then its
order is 1x n.
4 [4 w
049, Assertion (A): If ee rel (0 “I then
x=2y =2,z=-Sandwa4
Reason (R): Two matrices are equal, if their orders
are same and their corresponding elements are
equal.
Q60. Assertion (A): The product of two diagonal,
matrices of order 3 x 3is also a diagonal matrix.
Reason (R): Mat
non-commutative.
multiplication is always
QB1. Let A be a square matrix of order 3 satisfying
AM =I.
Assertion (A): A'= At.
Reason (R): (AB) = 8 A’.
52. Assertion (A): Let A=[a, ]be an mx n matrix and
be an mx n zero matrix, then A+0=0+A=A.
In other words, O is the additive identity for
matrix addition.
Reason (R): Let A=[4) ]n,cq be any matrix, then
we have another matrix as -A=[-0y Jn, such
that A+ (-A)=(-A)+A=0. Then, -A is the
additive inverse of A or negative of A.
(53. Assertion (A):For multiplication of two matrices A
and B, the number of columns in A should be less
than the number of rows in B.
Reason (R): For getting the elements of the
product matrix, we take rows of A and columns of
B, multiply them elementwise and take the sum.
Assertion (A): if A=[> 3} B=|> 9), th
984, Assertion (A): If A=], 4) B=|Q |, then
(A+B)? =A? +B? +248,
Reason (R): For the matrices A and 8 given in
Assertion (A), AB= BA.
tc? 2
Q85. Assertion (A): If A==/-2 1 2) then
2-2 -1
AAT) =1.
Reason (R): For any square matrix A, (A”)" =.
56. For any square matrix A with real number entries,
consider the following statements:
Assertion (A): A + 4’is a symmetric matrix.
Reason (R): AA’ is a skew-symmetric matrix.
