Topic: Matrices

One mark question ( V S A)

1. Define matrix

2. Define a diagonal matrix

3. Define scalar matrix

4. Define symmetric matrix

5. Define skew-symmetric matrix

6.In a matrix

find 1) order of the matrix

2) Write the elements of , , ,

7. If a matrix 8 elements what is the possible order it can have ?

8. If a matrix 18 elements what is the possible order it can have?

9. construct 2 matrix whose elements are given by

1) = 2) =

10. construct the 2 matrix whose elements are given by =

11. Construct the 3 matrix whose elements are given by =

12. Find x, y, z if =

13. Find x, y, z if =

14. Find the matrix x such that 2A + B + X =0 where A = and B =

15. If A = B= Find 2A – B

16. Find X if Y = and 2X+Y =

17. Find X If X+Y = and X-Y =

18. Simplify +

19. Find X If 2 + =

20. If A = Find A +

21. A = and B = Find 3A + 2B

22. if A = Verify A A1 = I

23. if B = verify B B1= I

24. If A = B= Find AB

25. Compute 1)

2)

26. If A square matrix of order 3 3 and = K Write the value of k

27. Find X and Y =

28. What is the number of possible square matrix order 3 with each entries 0 or 1

29. Find X and Y if is a scalar matrix

30. Find X is a symmetric matrix

II. Two mark and Three marks questions (SA)

1.Radha , fauzia, simran are the student of 12th class Radha has 15 note book and 6 pens ,
Fauzia has 10 books 2 pens and Simran has 13 books and 5 pens express this in to matrix
forms.

2. Construct 3 matrix whose elements are given by

3. Find X,Y,Z from the equation =

4. Find a,b,c, d From the equation ==

5. If A = B= Find X such that 2A + 3X = 5B

6. Find X and Y 2 + =

7. Find X and Y if x =

8. Given 3 = + Fine the values of X,Y,Z and W

9. If = and = Show that =

10. If A = and I = Find K If A2 = KA – 2I

11. If A = and B = verify (A + B )1 = A1 +B1

12. For any matrix A with real number entries , A+ A1 is symmetric matrix and A – A1
Skew-symmetric matrix

13. For any matrix A = verify that A+ A1 is symmetric matrix

14. For any matrix A = verify that A – A1 Skew-symmetric matrix

15. If A and B be the invertible matrices of same order then (AB)-1 = B-1A-1

16. By using elementary operation Find the inverse of the matrix

17. By using elementary operation Find the inverse of the matrix

18. By using elementary operation Find the inverse of the matrix

19. Find P-1 if it exists and P =

20. If A = Show that A2 -5A +7I = 0

21. If A = then prove that An = where n is any positive integer

22. If A and B are symmetric matrices of same order show that AB is symmetric if and
only if AB = BA

23. I f A = find A2

24. If A = and B = Show that (AB)1 = B1A1

25. Find X, Y, Z if

III. Five mark questions ( LA)

1.If A = B= and C =

Find A B , BC and show that (AB )C = A(BC)

2. If A = B= C= calculate AC, BC and (A+B) C

Deduce that (A+B) C = AC + BC

3. If A = Show that A3 – 23A - 40 I = 0

4. If A = B= and C =

verify A+ (B-C) = (A+B ) –C

5. If A = and B = find 3A – 5B
6. If A = find A2 – 5 A + 6 I ?

7. If A = prove that A3 – 6A2 + 7A + 2I = 0

8. Express the matrix B = Find the sum of symmetric and skew-

symmetric matrix

9. Express the matrix B = Find the sum of symmetric and skew-

symmetric matrix

10. Find the inverse of the matrix using elementary operation A =

11. Find the inverse of the matrix using elementary operation A =

12. If A = B= C= calculate AB , BC, A(B+C)

Verify that AB + AC = A(B+C)

13. If F(x) = show that F(x) F(y) = F(x+y)

14. If A = and B = verify (AB)1 = B1A1

15. If A = Prove that An =

 Solutions
One mark questions (VSA)
1. The numbers arranged in rectangular array of rows and columns by the brockets is
called matrix
2. A square matrix is said to be diagonal matrix if all non diagonals elements are zeros
3. A diagonal matrix is said to be scalar marics if it’s diagonal elements are equal
4. If a square matrix A = is said to be symmetric if and only if A1 = A
5. If a square matrix A = is said to be skew-symmetric if and only if A1 = -A
6. 1) order of the matrix is 2) 19, -2, -5 , 12,
7. Possible orders are (1,8) (8,1) (2,4) (4,2) is 1X8 , 8X 1 , 2X 4 , 4X 2
8. Possible orders are (1,18) (18,1) (3,6) (6,3) (2,9) (9,2) is 1 X18 , 18X1 , 3X6 , 6X3 ,
2X9, 9X2 ,

9. 1) = 2)

10.

11.

12. X= 1 Y=4 Z= 3
13. X = 2 Y=4 Z=0
14. X = -2A - B = - =

15. 2A –B =

16. By solving X =

17. By solving above matrix X = and Y =

18. By multiplying we get the answer =I

19. 2+Y = 5 implies Y = 3 and 2x+2 = 8 implies x =3
20. A + A1 =

21. 3 A +2 B =
 22. A A1 = = = I after multiplying

23. BB 1 = = = I after multiplying

24. (AB)1 =

25. 1) 2) after multiplying

26. 3 K implies K = 3
27. Y = 0, X = 3 by solving
28. The square matrix of order 3X3 = 9 and 2 entries

Then possible entries is 29 = 512

29. = then X = 2 Y = 4

30. = implies X = 5

Solutions : Two mark and Three marks questions (SA)

1. books pens

Radha : 15 6 this can be expressed as or

Fauzia : 10 2
Simran: 13 5

2. =

3. X+ Y + Z = 9 X +Z = 5 Y+Z=7

7+Z=9 X+2=5 Y+2=7

Z= 2 X=3 Y=5

4. By solving equality a =1 , b= 2, c =3 and d = 4

5. X =
6. compare two matrices X = 2, Y = 9

7. by solving we get X = 3 , Y = -4

8. by solving and compare we get X = 2 , Y = 4, Z = 1, w=3

9. = = =

10. A2 = KA – 2I

Then 4K = 4

K =1

11. ( A+B)1 = and A1 + B 1 =

Hence ( A+B)1 = A1 + B 1

12. B = A + A1 , B1 = (A + A1 )1 = A1+A = B B = A +A1 is symmetric

C = A –A 1 , C 1 = (A -A1 )1 = A1-A = -(A- A1) = - C C = A – A1 is skew- symmetric

13. Z = A + A1 = + = = Z1 Z = Z1 = A + A1 is symmetric

14. Z1 = (A - A1)1 = = = -Z Z1 = - Z, A –A1

skew- symmetric
15. (AB) (AB)-1 = I
A-1(AB) (AB)-1 = A-1I IA=A
B(AB)-1 = A-1 IA-1 = A-1
B-1B(AB)-1 = B-1A-1 AA-1 = I
(AB)-1 = B-1A-1 BB-1 I
16. A=
A = IA
= A
 = A

= A

= A

A -1 =

17. By above process A -1 =

18. By above prose’s A -1 =

19. P =
P = IP
= P
By elementary operation

= p

p-1 does not exists

20. A2 -5A +7I = - + =0
21. By mathematical induction we get the solution
22. If A=A1 , B = B1 , (AB)1 = AB
(AB)1 = B1 A1 = BA AB = BA AB Is symmetric
2
23. A = A A By product of two matrix get the solution
24. (AB)1 =

B1A1 =
(AB)1 = B1A1
25. By solving x = 2 , y = 4 , z = 3
Solutions : Five mark questions (LA)
1.AB = (AB) C =
 BC = A(BC) =

Hence (AB) C = A(BC)

2. (A +B) C = AC = BC = AC + BC =

Hence (A +B) C = AC + BC

3. A = A2 = A3 =

LHS = A3 – 23A – 40 I = 0 By simplification

4. A + (B – C) = and (A+B) –C =

Hence A + (B – C) = (A+B) –C

5. 3A -5B = - = =0

6. A2 – 5A + 6I = by simplification

7. If A = by calculating A2 , A3 take LHS = RHS

8. B = by theorem number 2

B = (B +B 1) + (B -B 1) hence they are equal

9. B = by theorem number 2

B = (B +B 1) + (B -B 1) hence they are equal

10. if A = by elementary operation we get inverse of the matrix

11. If A = by elementary operation we get inverse of the matrix

12. If AB = AC = A(B+C) = = AB + AC

13. F(x).F(y) =

= = F(x+y)

14. LHS = (AB)1 = = B1A1 = RHS

15. By mathematical induction we get the solution