# RD Sharma Class 12 Ex 5.4 Solutions Chapter 5 Algebra of Matrices

Here we provide RD Sharma Class 12 Ex 5.4 Solutions Chapter 5 Algebra of Matrices for English medium students, Which will very helpful for every student in their exams. Students can download the RD Sharma Class 12 Ex 5.4 Solutions Chapter 5 Algebra of Matrices book pdf download. Now you will get step-by-step solutions to each question.

## RD Sharma Class 12 Ex 5.4 Solutions Chapter 5 Algebra of Matrices

### Question 1: Let A = and B = verify that

(i) (2A)T = 2AT

(ii) (A + B)T = AT + BT

(iii) (A − B)T = AT − BT

(iv) (AB)T = BT AT

Solution:

(i) Given: A = and B = Assume,

(2A)T = 2AT

Substitute the value of A

L.H.S = R.H.S

Hence, proved.

(ii) Given: A = and B = Assume,

(A+B)T = AT + BT

L.H.S = R.H.S

Hence, proved.

(iii) Given: A= and B= Assume,

(A − B)T = AT − BT

L.H.S = R.H.S

Hence, proved

(iv) Given: A = and B = Assume,

(AB)T = BTAT

Therefore, (AB)T = BTAT

Hence, proved.

### Question 2: A = and B = Verify that (AB)T = BTAT

Solution:

Given: A = and B = Assume,

(AB)T = BTAT

L.H.S = R.H.S

Hence proved

### Find AT, BT and verify that

(i) (A + B)T = AT + BT

(ii) (AB)T = BTAT

(iii) (2A)T = 2AT

Solution:

(i) Given: A = and B = Assume

(A + B)T = AT + BT

L.H.S = R.H.S

Hence proved

(ii) Given: A = and B = Assume,

(AB)T = BTAT

L.H.S =R.H.S

Hence proved

(iii) Given: A = and B = Assume,

(2A)T = 2AT

L.H.S = R.H.S

Hence proved

### Question 4: if A = , B = , verify that (AB)T = BTAT

Solution:

Given: A = and B = Assume,

(AB)T = BTAT

L.H.S = R.H.S

Hence proved

### Question 5: If A = and B = , find (AB)T

Solution:

Given: A = and B = Here we have to find (AB)T

Hence,

(AB)T = ### (i) For two matrices A and B, verify that (AB)T = BTAT

Solution:

Given,

(AB)T = BTAT

⇒ ⇒ ⇒ ⇒ ⇒ L.H.S = R.H.S

Hence,

(AB)T = BTAT

### (ii) For the matrices A and B, verify that (AB)T = BTAT, where

Solution:

Given,

(AB)T = BTAT

⇒ ⇒ ⇒ ⇒ ⇒ L.H.S = R.H.s

So,

(AB)T = BTAT

### Question 7: Find , AT – BT

Solution:

Given that We need to find AT – BT.

Given that, Let us find AT – BT

⇒ ⇒ ⇒ ### Question 8: If , then verify that A’A = 1

Solution:

⇒ ⇒ ⇒ Hence,we have verified that A’A = I

### Question 9: , then verify that A’A = I

Solution:

Hence, we have verified that A’A = I

### Where Solution:

Given,

li, mi, ni are direction cosines of three mutually perpendicular vectors

⇒ And,

Given,

= I

Hence,

AAT = I

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