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# Solutions 3. Matrices - Exercise 3.3 | Class 12 Mathematics-I - Toppers Study

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## Solutions 3. Matrices - Exercise 3.3 | Class 12 Mathematics-I - Toppers Study

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### You can Find Mathematics-I solution Class 12 Chapter 3. Matrices

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Chapter 3 Mathematics-I class 12

### Exercise 3.3 class 12 Mathematics-I Chapter 3. Matrices

Sure! The following topics will be covered in this article

- Solutions 3. Matrices - Exercise 3.3 | Class 12 Mathematics-I - Toppers Study
- Class 12 Ncert Solutions
- Solution Chapter 3. Matrices Class 12
- Solutions Class 12
- Chapter 3. Matrices Exercise 3.3 Class 12

## 3. Matrices

### | Exercise 3.3 |

## Solutions 3. Matrices - Exercise 3.3 | Class 12 Mathematics-I - Toppers Study

**Exercise 3.3**

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

**(i) **

**(ii) **

**(iii) **

**Ans. (i)** Let A =

Transpose of A = A’ or AT =

**(ii)**

Transpose of A = A’ or AT =

**(iii)**

Transpose of A = A’ or AT =

**Ques.2. If A = and B = then verify that:**

**(i) **

**(ii) **

**Ans. (i)** A + B = = =

L.H.S. = (A + B)’ = =

R.H.S. = A’ + B’ =

=

= =

L.H.S. = R.H.S.

hance Proved.

**(ii)** A – B =

= =

L.H.S. = (A – B)’ = =

R.H.S. = A’ – B’ =

=

= =

L.H.S. = R.H.S.

hance Proved.

**Ques.3. If A’ = and B = then verify that:**

**(i) **

**(ii) **

**Ans. **Given: A’ = and B = then (A’)’ = A =

**(i)** A + B =

=

L.H.S. = (A + B)’ =

R.H.S. = A’ + B’

=

=

= =

L.H.S. = R.H.S.

hance Proved.

**(ii)** A – B =

=

L.H.S. = (A – B)’

=

R.H.S. = A’ – B’

=

=

= =

L.H.S. = R.H.S.

hance Proved.

**Ques.4. If A’ = and B = then find (A + 2B)’.**

**Ans.**** **Given: A’ = and B = then (A’)’ = A =

A +2B =

= ]

= =

(A + 2B)’ =

**Ques.5. For the matrices A and B, verify that (AB)’ = B’A’, where:**

**(i) A = B = **

**(ii) A = B = **

**Ans. (i)** AB = =

L.H.S. = (AB)’

= =

R.H.S. = B’A’

=

=

=

L.H.S. = R.H.S.

hance Proved.

**(ii)** AB = =

L.H.S. = (AB)’

=

=

R.H.S. = B’A’

=

=

=

L.H.S. = R.H.S.

hance Proved.

**Ques.6. (i) If A = then verify that A’A = I.**

**(ii) If A = then verify that A’A = I.**

**Ans. (i)** L.H.S. = A’A =

=

= = = I = R.H.S.

**(ii)** L.H.S. = A’A

=

=

= =

= I = R.H.S.

**Ques.7. (i) Show that the matrix A = is a symmetric matrix.**

**(ii) Show that the matrix A = is a skew symmetric matrix.**

**Ans. (i)** Given: A = ……….(i)

Changing rows of matrix A as the columns of new matrix A’ = = A

A’ = A

hance, by definitions of symmetric matrix, A is a symmetric matrix.

**(ii)** Given: A = ……….(i)

A’ =

=

Taking common, A’ = = – A [From eq. (i)]

hance, by definition matrix A is a skew-symmetric matrix

**Ques.8. For a matrix A = verify that:**

**(i) (A + A’) is a symmetric matrix.**

**(ii) (A – A’) is a skew symmetric matrix.**

**Ans. (i)** Given: A =

Let B = A + A’ =

=

=

B’ = = B

B = A + A’ is a symmetric matrix.

**(ii)** Given:

Let B = A – A’ =

=

=

B’ =

Taking common, = – B

B = A – A’ is a skew-symmetric matrix.

**Ques.9. Find (A + A’) and (A – A’) when A = **

**Ans. **Given: A = A’ =

Now, A + A’ = = =

(A + A’) =

Now, A – A’ = = =

(A – A’) = =

**Ques.****10****. Express the following matrices as the sum of a symmetric and skew symmetric matrix:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Ans. (i)** Given: A = so, A’ =

Symmetric matrix = (A + A’)

=

= =

Skew symmetric matrix = (A – A’)

=

= =

Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .

**(ii)** Given: A = so, A’ =

Symmetric matrix = (A + A’)

=

= =

And Skew symmetric matrix = (A – A’)

=

= =

Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .

**(iii)** Given: A = so, A’ =

Symmetric matrix = (A + A’)

=

= =

And Skew symmetric matrix = (A–A’)

=

= =

Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .

**(iv)** Given: A = so, A’ =

Symmetric matrix = (A + A’)

=

= =

And Skew symmetric matrix = (A – A’)

=

=

Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .

**Ques.11. If A and B are symmetric matrices of same order, AB – BA is a:**

**(A) Skew-symmetric matrix**

**(B) Symmetric matrix**

**(C) Zero matrix**

**(S) Identity matrix**

**Ans. **Given: A and B are symmetric matrices A = A’ and B = B’

Now, (AB – BA)’ = (AB)’ – (BA)’

(AB – BA)’ = B’A’ – A’B’ [Reversal law]

(AB – BA)’ = BA – AB [From eq. (i)]

(AB – BA)’ = – (AB – BA)

(AB – BA) is a skew matrix.

hance, option (A) is correct.

**Ques.12. If A = , then A + A’ = I, if the value of is:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Given: A = Also A + A’ = I

Equating corresponding entries, we have

hance, option (B) is correct.

##### Other Pages of this Chapter: 3. Matrices

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