## NCERT Solutions for Class 12 Maths Exercise 3.3 NCERT Solutions for Class 12 Maths Chapter 3 â€“ Matrices â€“ FREE PDF Download

NCERT Class 12 Maths Ch 3 is one of the most important ones in the NCERT syllabus. Duly following NCERT Solutions for Class 12 Maths Chapter 3 ensures you that there will be no hindrance when you opt for more advanced branches of Maths. Class 12 Maths Chapter 3 â€“ Matrices solved by Expert Teachers as per NCERT (CBSE) Book guidelines. All Matrices Exercise Questions with Solutions to help you to revise complete Syllabus and Score More marks.

# NCERT Solutions for Class 12 Maths Chapter 3 â€“ Matrices

(i)Â

(ii)Â

(iii)Â

**Ans.**Â (i)Â Let A =Â

Â Transpose of A = Aâ€™ or A

^{T}Â =Â

(ii)Â

Â Transpose of A = Aâ€™ or A

^{T}Â =Â

(iii)Â

Â Transpose of A = Aâ€™ or A

^{T}Â =Â

### 2. If A =Â Â and B =Â Â then verify that:

(i)Â

(ii)Â

**Ans.**Â (i)Â A + B =Â Â =Â Â =Â

L.H.S. = (A + B)â€™ =Â Â =Â

R.H.S. = Aâ€™ + Bâ€™ =Â Â =Â

=Â Â =Â

Â L.H.S. = R.H.S.Â Â Â Â Â Â Â Â Proved.

(ii)Â A â€“ B =Â Â =Â Â =Â

L.H.S. = (A â€“ B)â€™ =Â Â =Â

R.H.S. = Aâ€™ â€“ Bâ€™ =Â Â =Â

=Â Â =Â

Â L.H.S. = R.H.S.Â Â Â Â Â Â Â Â Proved.

### 3. If Aâ€™ =Â Â and B =Â Â then verify that:

(i)Â

(ii)Â

**Ans.**Â Given: Aâ€™ =Â Â and B =Â Â then (Aâ€™)â€™ = A =Â

(i)Â A + B =Â Â =Â

Â L.H.S. = (A + B)â€™ =Â

R.H.S. = Aâ€™ + Bâ€™ =Â Â =Â

=Â Â =Â

Â L.H.S. = R.H.S.Â Â Â Â Â Â Â Â Proved.

(ii)Â A â€“ B =Â Â =Â

Â L.H.S. = (A â€“ B)â€™ =Â

R.H.S. = Aâ€™ â€“ Bâ€™ =Â Â =Â

=Â Â =Â

Â L.H.S. = R.H.S.Â Â Â Â Â Â Â Â Proved.

### 4. If Aâ€™ =Â Â and B =Â Â then find (A + 2B)â€™.

**Ans.**Â Given: Aâ€™ =Â Â and B =Â Â then (Aâ€™)â€™ = A =Â

A +2B =Â

=Â

=Â

=Â

Â (A + 2B)â€™ =Â

### 5. For the matrices A and B, verify that (AB)â€™ = Bâ€™Aâ€™, where:

(i) A =Â Â B =Â

(ii) A =Â Â B =Â

**Ans.**Â (i)Â AB =Â Â =Â

Â L.H.S. = (AB)â€™ =Â Â =Â

R.H.S. = Bâ€™Aâ€™ =Â Â =Â Â =Â

Â L.H.S. = R.H.S.Â Â Â Â Â Â Â Â Proved.

(ii)Â AB =Â Â =Â

Â L.H.S. = (AB)â€™ =Â Â =Â

R.H.S. = Bâ€™Aâ€™ =Â Â =Â Â =Â

Â L.H.S. = R.H.S.Â Â Â Â Â Â Â Â Proved.

### 6. (i) If A =Â Â then verify that Aâ€™A = I.

(ii) If A =Â Â then verify that Aâ€™A = I.

**Ans.**Â (i)Â L.H.S. = Aâ€™A =Â

=Â

=Â Â =Â Â = I = R.H.S.

(ii)Â L.H.S. = Aâ€™A =Â Â =Â

=Â Â =Â Â = I = R.H.S.

### 7. (i) Show that the matrix A =Â Â is a symmetric matrix.

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

**Ans.**Â (i)Â Given: A =Â Â Â â€¦â€¦â€¦.(i)

Changing rows of matrix A as the columns of new matrix Aâ€™ =Â Â = A

Â Aâ€™ = A

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

(ii)Â Given: A =Â Â â€¦â€¦â€¦.(i)

Â Aâ€™ =Â Â =Â

TakingÂ Â common, Aâ€™ =Â Â = â€“ AÂ Â [From eq. (i)]

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

### 8. For a matrix A =Â Â verify that:

(i) (A + Aâ€™) is a symmetric matrix.

(ii) (A â€“ Aâ€™) is a skew symmetric matrix.

**Ans.**Â (i)Â Given: A =Â

Let B = A + Aâ€™ =Â Â =Â Â =Â

Â Bâ€™ =Â Â = B

Â B = A + Aâ€™ is a symmetric matrix.

(ii)Â Given:Â

Let B = A â€“ Aâ€™ =Â Â =Â Â =Â

Â Bâ€™ =Â

TakingÂ Â common,Â = â€“ B

Â B = A â€“ Aâ€™ is a skew-symmetric matrix.

### 9. FindÂ Â (A + Aâ€™) andÂ Â (A â€“ Aâ€™) when A =Â

**Ans.**Â Given: A =Â

Â Â Aâ€™ =Â

Now, A + Aâ€™ =Â

=Â

=Â

Â Â Â (A + Aâ€™) =Â

Now, A â€“ Aâ€™ =Â

=Â

=Â

Â Â Â (A â€“ Aâ€™) =Â

=Â

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

(i)Â

(ii)Â

(iii)Â

(iv)Â

**Ans.**Â (i)Â Given: Â A =Â Â Â Â Â Â Â Aâ€™ =Â

Â Symmetric matrix =Â Â (A + Aâ€™) =Â

=Â Â =Â

And Skew symmetric matrix =Â Â (A â€“ Aâ€™) =Â

=Â Â =Â

=> A =Â Â +Â .

(ii)Â Given: A =Â Â Â Â Â Â Aâ€™ =Â

Â Symmetric matrix =Â Â (A + Aâ€™) =Â

=Â Â =Â

And Skew symmetric matrix =Â Â (A â€“ Aâ€™) =Â

=Â Â =Â

A =Â Â +Â .

(iii)Â Given: A =Â Â Â Â Â Â Aâ€™ =Â

Â Symmetric matrix =Â Â (A + Aâ€™) =Â

=Â Â =Â

And Skew symmetric matrix =Â Â (A â€“ Aâ€™) =Â

=Â Â =Â

A = Â Â +Â .

(iv)Â Given: A =Â Â Â Â Â Â Â Aâ€™ =Â

Â Symmetric matrix =Â Â (A + Aâ€™) =Â Â =Â Â =Â

And Skew symmetric matrix =Â Â (A â€“ Aâ€™) =Â Â =Â

A = Â +Â .

### Choose the correct answer in Exercises 11 and 12.

11. If A and B are symmetric matrices of same order, AB â€“ BA is a:

(A) Skew-symmetric matrix

(B) Symmetric matrix

(C) Zero matrix

(S) Identity matrix

**Ans.**Â Given: A and B are symmetric matricesÂ Â Â A = Aâ€™ and B = Bâ€™

Now, (AB â€“ BA)â€™ = (AB)â€™ â€“ (BA)â€™

Â (AB â€“ BA)â€™ = Bâ€™Aâ€™ â€“ Aâ€™Bâ€™ [Reversal law]

Â (AB â€“ BA)â€™ = BA â€“ AB [From eq. (i)]

Â (AB â€“ BA)â€™ = â€“ (AB â€“ BA)

Â (AB â€“ BA) is a skew matrix.

Therefore, option (A) is correct.

### 12. If A =Â , then A + Aâ€™ = I, if the value ofÂ Â is:

(A)Â

(B)Â

(C)Â

(D)Â

**Ans.**Â Given: A =Â Â Â Also A + Aâ€™ = I

Â Â Â Â Â |

Equating corresponding entries, we have

Â Â Â Â Â Â Â Â Â

Therefore, option (B) is correct.