Study Materials: ncert solutions

Our ncert solutions for Solutions 3. Matrices - Exercise 3.3 | Class 12 Mathematics-I - Toppers Study is the best material for English Medium students cbse board and other state boards students.

Solutions ⇒ Class 12th ⇒ Mathematics-I ⇒ 3. Matrices

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

Topper Study classes prepares ncert solutions on practical base problems and comes out with the best result that helps the students and teachers as well as tutors and so many ecademic coaching classes that they need in practical life. Our ncert solutions for Solutions 3. Matrices - Exercise 3.3 | Class 12 Mathematics-I - Toppers Study is the best material for English Medium students cbse board and other state boards students.

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

CBSE board students who preparing for class 12 ncert solutions maths and Mathematics-I solved exercise chapter 3. Matrices available and this helps in upcoming exams 2023-2024.

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

• All Chapter review quick revision notes for chapter 3. Matrices Class 12
• NCERT Solutions And Textual questions Answers Class 12 Mathematics-I
• Extra NCERT Book questions Answers Class 12 Mathematics-I
• Importatnt key points with additional Assignment and questions bank solved.

Chapter 3 Mathematics-I class 12

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

• 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

## 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

Important Study materials for classes 06, 07, 08,09,10, 11 and 12. Like CBSE Notes, Notes for Science, Notes for maths, Notes for Social Science, Notes for Accountancy, Notes for Economics, Notes for political Science, Noes for History, Notes For Bussiness Study, Physical Educations, Sample Papers, Test Papers, Mock Test Papers, Support Materials and Books.

Mathematics Class - 11th

NCERT Maths book for CBSE Students.

books

## Study Materials List:

##### Solutions ⇒ Class 12th ⇒ Mathematics-I
1. Relations and Functions
2. Inverse Trigonometric Functions
3. Matrices
4. Determinants
5. Continuity And Differentiability
6. Application of Derivatives

New Books