## Rajasthan Board RBSE Class 12 Maths Chapter 3 Matrix Miscellaneous Exercise

Question 1.

, then find A.

Solution:

Question 2.

, then find (A – 2I) (A – 3I).

Solution:

Question 3.

, then find AB.

Solution:

Question 4.

, then find BA.

Solution:

Question 5.

, then find matrices A and B.

Solution:

Question 6.

, then find the value of x and y.

Solution:

On comparing,

x + 2 = -2 ∴ x = -4

– y – 2 = 5 ⇒ y = -7

Hence, x = -4, y = -7

Question 7.

Order of matrix A is 3 x 4 and B is a matrix, such that A^{T}B and AB^{T} defined, then write the order of B.

Solution:

∴ Order of A = 3 x 4

∴ Order of A^{T} = 4 x 3

But A^{T}B and AB^{T} is defined.

So, order of B is 3 x 4.

Question 8.

, is a symmetric matrix, then determine x.

Solution:

Given,

On comparing a_{ij} = a_{ji}

a_{32} = a_{23} ⇒ -x = 4

∴ x = -4

Question 9.

Construct a matrix of order 3 x 3, B = [b_{ij}], whose elements are b_{ij}= (i) (j).

Solution:

B_{11} = 1 x 1 = 1

B_{12} = 1 x 2 = 2

B_{13} = 1 x 3 = 3

B_{21} = 2 x 1 = 2

B_{22} = 2 x 2 = 4

B_{23} = 2 x 3 = 6

B_{31} = 3 x 1 = 3

B_{32} = 3 x 2 = 6

B_{33} = 3 x 3 = 9

Question 10.

Solution:

Question 11.

Express matrix A as the sum of symmetric and skew-symmetric matrices, where

.

Solution:

Given,

Question 12.

then prove that :

(i) (A^{T})^{T} = A

(ii) A + A^{T} is a symmetric matrix.

(iii) A – A^{T} is a skew-symmetric matrix.

(iv) AA^{T} and A^{T}A are symmetric matrix.

Solution:

(i) (A^{T})^{T} = A

(ii) A + A^{T} is symmetric

So, A + A^{T} is symmetric matrix. Hence Proved.

So, A – A^{T} is skew symmetric matrix. Proved.

Here,

a_{21} = a_{12} = 0

a_{31} = a_{13} = 0

a_{23} = a_{32} = 6

So, AA^{T} is symmetric matrix.

Here,

a_{12} = a_{21} = 0

a_{13} = a_{31} = 0

a_{32} = a_{23} = 4

So, A^{T}A is symmetric matrix.

Question 13.

, and 3A – 2B + C is a null matrix, then determine matrix ‘C’.

Solution:

Question 14.

Construct a matrix B = [b_{ij}] of the order 2 x 3, whose elements are b_{ij} = (i +2j)^{2}/2

Solution:

Given, B = [b_{ij}] whose elements are

Question 15.

, then find the element of 1^{st} row of ABC.

Solution:

So, element of 1^{st} row is 8.

Question 16.

, then find AA^{T}.

Solution:

Given,

Question 17.

, then find x.

Solution:

Question 18.

, then prove

Solution:

Given,

= (bc – ad)I_{2} = R.H.S.

Hence Proved.

Question 19.

, then find the matrix form of the following (aA + bB) (aA – bB).

Solution:

Givn,

Question 20.

, then prove that (A – B)^{2} ≠ A^{2} – 2AB + B^{2}.

Solution:

Given,

From (i) and (ii),

(A – B)^{2} + A^{2} – 2AB + B^{2}

Hence Proved.

Question 21.

, then find k, where A^{2} = kA – 2IA_{2} .

Solution:

Given,

On comparing corresponding element

From 3k – 2 = 1

3k = 3 ⇒ k = ⇒ k = 1.

Question 22.

i = √-1 then prove that :

(1) A^{2} = B^{2} = C^{2} = -I_{2}

(ii) AB = – BA = -C

Solution:

Question 23.

and f(A) = A^{2} – 5A + 7I then find f(A).

Solution:

Question 24.

Prove that

Solution:

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