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NCERT Solutions for Class 12

Maths

Chapter 3 - Matrix

Exercise 3.3

1. Find the transpose of each of the following matrices:

(i)

5

1 / 2

-1

Ans: Find transpose

T

51

A 1/ 2 A 5 12

-1

(ii) 1 -1

2 3

Ans: Find transpose

T1 -1 1 2

A A2 3 1 3

(iii)

-1 5 6

3 5 6

2 3 -1

Ans: Find transpose

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T

-1 5 6 -1 3 2

A 3 5 6 A 5 5 3

2 3 -1 6 6 -1

2. If

-1 2 3

A = 5 7 9

-2 1 1

and

-4 1 -5

B = 1 2 0

1 3 1

, then verify that

(i) A + B ' = A' + B'

(ii) A - B ' = A' - B'

Ans: We have

-1 5 -2 -4 1 1

A' 2 7 1 B' 1 2 3

3 9 1 -5 0 1

(i) A + B ' = A' + B'

Ans:

5 3 2 5 6 1

A B 6 9 9 A B ' 3 9 4

1 4 2 2 9 2

-1 5 -2 -4 1 1 -5 6 1

A' B' 2 7 1 1 2 3 3 9 4

3 9 1 -5 0 1 -2 9 2

Hence, we have verified that A B ' A' B'

(ii) A - B ' = A' - B'

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Ans:

3 1 8 3 4 3

A B 4 5 9 A B ' 1 5 2

3 2 0 8 9 0

-1 5 -2 -4 1 1 3 4 3

A' B' 2 7 1 1 2 3 1 5 2

3 9 1 -5 0 1 8 9 0

Hence, we have verified that A - B ' A'- B'

3. If

3 4

A' = -1 2

0 1

and -1 2 1

B =1 2 3

, then verify that

(i) A + B ' = A' + B'

(ii) A - B ' = A' - B'

Ans: We have

1 13 -1 0

A B' 2 24 2 1

1 3

(i) A + B ' = A' + B'

Ans:

2 52 1 1

A B A B ' 1 45 4 4

1 4

3 4 1 1 2 5

A' B' 1 2 2 2 1 4

0 1 1 3 1 4

Hence, we have verified that A B ' A' B'

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(ii) A - B ' = A' - B'

Ans:

4 34 3 1

A B A B ' 3 03 0 2

1 2

3 4 1 1 4 3

A' B' 1 2 2 2 3 0

0 1 1 3 1 2

Hence, we have verified that A - B ' A'- B'

4. If -2 3 -1 0

A' = & B =1 2 1 2

, then find A + 2B '

Ans: Solve for the condition

2 1A

3 2

2 1 1 0 2 1 2 0 4 1A 2B 2

3 2 1 2 3 2 2 4 5 6

4 5(A 2B)

1 6

5. For the matrix A & B , verify that AB ' = B'A'where

1

(i) A = -4 ,B = -1 2 1

3

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Ans:

1 1 2 1

AB 4 1 2 1 4 8 4

3 3 6 3

1 4 3

(AB) 2 8 6

1 4 3

1 1 4 3

B'A' 2 1 4 3 2 8 6

1 1 4 3

Hence, we have verified AB ' B'A'

0

(ii)A = 1 ,B = 1 5 7

2

Ans:

0 0 0 0

AB 1 1 5 7 1 5 7

2 2 10 14

0 1 2

(AB) 0 5 10

0 7 14

1 0 1 2

B'A' 5 0 1 2 0 5 10

7 0 7 14

Hence, we have verified AB ' B'A'

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6. If

cosα sinα(i) A = , then verify thatA'A = I

-sinα cosα

Ans: cos sin

Asin cos

cos sinA'

sin cos

cos sin cos sinA'A

sin cos sin cos

2 2

2 2

cos sin sin cos sin cosA'A

sin cos sin cos sin cos

1 0A'A I

0 1

sinα cosα(ii) A = , then verify thatA'A = I

-cosα sinα

Ans: sin cos

Acos sin

2 2

2 2

sin cosA'

cos sin

sin cos sin cosA'A

cos sin cos sin

cos sin sin cos sin cosA'A

sin cos sin cos sin cos

1 0A'A I

0 1

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7.

(i) Show that the matrix

1 -1 5

A = -1 2 1

5 1 3

is a symmetric matrix

Ans: We have

1 -1 5

A' -1 2 1 A

5 1 3

A' A

Hence, A is a symmetric matrix

(ii) Show that the matrix

0 1 -1

A = -1 0 1

1 -1 0

is a skew symmetric matrix

Ans: We have

0 1 1 0 1 -1

A' 1 0 1 -1 0 1 A

1 1 0 1 -1 0

A' A

Hence, A is a skew-symmetric matrix

8. For the matrix 1 5

A =6 7

, verify that

(i) A + A' is a symmetric matrix

(ii) A + A' is a skew symmetric matrix

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Ans: 1 5 1 6

A ,A'6 7 5 7

(i) A + A' is a symmetric matrix

Ans: 1 5 1 6 2 11

A A'6 7 5 7 11 14

2 11

A A' ' A A'11 14

Hence, A A' is a symmetric matrix

(ii) A + A' is a skew symmetric matrix

Ans: 1 5 1 6 0 1

A A'6 7 5 7 1 0

0 1 0 1

A A' ' A A'1 0 1 0

Hence, A A' is a skew-symmetric matrix

9. Find 1 1

A + A' & A - A'2 2

, when

0 a b

A = -a 0 c

-b -c 0

Ans: The given matrix is

0 a b

A -a 0 c

-b -c 0

, then

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0 a b

A' a 0 c

b c 0

0 a b 0 a b1 1

A A' -a 0 c a 0 c2 2

-b -c 0 b c 0

0 0 01

A A' 0 0 02

0 0 0

0 a b 0 a b1 1

A A' -a 0 c a 0 c2 2

-b -c 0 b c 0

0 a b1

A A' -a 0 c2

-b -c 0

10. Express the following matrices as the sum of a symmetric and a skew

symmetric matrix:

3 5

1 -1

Ans: 3 5 3 11 1

A A'1 -1 5 -12 2

6 61 1A A'

6 22 2

3 31A A'

3 -12

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Thus, 1

A A'2

is a symmetric matrix.

3 5 3 11 1A A'

1 -1 5 -12 2

0 41 1A A'

4 02 2

0 21A A'

2 02

Thus, 1

A A'2

is a skew-symmetric matrix.

11. If A,B are symmetric matrix of same order, then AB - BA is a

A. Skew symmetric matrix

B. Symmetric matrix

C. Zero matrix

D. Identity matrix

Ans: The correct answer is A

A & B are symmetric , therefore , we have

A' A & B' B

Consider

AB BA ' AB ' BA '

B'A ' A 'B'

BA AB

AB BA

AB BA ' AB BA

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Thus, AB BA ' is a skew-symmetric matrix

12. If cosα -sinα

A =sinα cosα

, then A + A' = I , if the value of α is

A. π

6

B. π

3

C. π

D. 3π

2

Ans: The correct answer is B

cos -sinA

sin cos

cos sinA'

sin cos

A A' I

cos -sin cos sin 1 0

sin cos sin cos 0 1

2cos 0 1 0

0 2cos 0 1

3