# Class 12 RD Sharma Solutions- Chapter 5 Algebra of Matrices – Exercise 5.4

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

**(i) (2A) ^{T} = 2A^{T}**

**(ii) (A + B) ^{T} = A^{T} + B^{T}**

**(iii) (A − B) ^{T} = A^{T} − B^{T}**

**(iv) (AB) ^{T} = B^{T} A^{T}**

**Solution:**

(i)Given: A = and B =Assume,

(2A)

^{T}= 2A^{T}Substitute the value of A

L.H.S = R.H.S

Hence, proved.

(ii)Given: A = and B =Assume,

(A+B)

^{T}= A^{T}+ B^{T}L.H.S = R.H.S

Hence, proved.

(iii)Given: A= and B=Assume,

(A − B)

^{T}= A^{T}− B^{T}L.H.S = R.H.S

Hence, proved

(iv)Given: A = and B =Assume,

(AB)

^{T}= B^{T}A^{T}Therefore, (AB)

^{T}= B^{T}A^{T}Hence, proved.

**Question 2: A =** **and B = **** Verify that (AB)**^{T} = B^{T}A^{T}

^{T}= B

^{T}A

^{T}

**Solution**:

Given: A = and B =

Assume,

(AB)

^{T}= B^{T}A^{T}L.H.S = R.H.S

Hence proved

**Question 3: Let A = ****and B = **

**Find A**^{T}, B^{T} and verify that

^{T}, B

^{T}and verify that

**(i) (A + B) ^{T} = A^{T} + B^{T}**

**(ii) (AB) ^{T} = B^{T}A^{T}**

**(iii) (2A) ^{T} = 2A^{T}**

**Solution:**

(i)Given: A =and B =

Assume

(A + B)

^{T}= A^{T}+ B^{T}L.H.S = R.H.S

Hence proved

(ii)Given: A = and B =Assume,

(AB)

^{T}= B^{T}A^{T}L.H.S =R.H.S

Hence proved

(iii)Given: A = and B =Assume,

(2A)

^{T}= 2A^{T}L.H.S = R.H.S

Hence proved

**Question 4: if A = ****, B = ****, verify that (AB)**^{T} = B^{T}A^{T}

^{T}= B

^{T}A

^{T}

**Solution:**

Given: A = and B =

Assume,

(AB)

^{T}= B^{T}A^{T}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}=

**Question 6: **

**(i) For two matrices A and B, **** verify that (AB)**^{T} = B^{T}A^{T}

^{T}= B

^{T}A

^{T}

**Solution:**

Given,

(AB)

^{T}= B^{T}A^{T}⇒

⇒

⇒

⇒

⇒ L.H.S = R.H.S

Hence,

(AB)

^{T}= B^{T}A^{T}

**(ii) For the matrices A and B, verify that (AB)**^{T} = B^{T}A^{T}, where

^{T}= B

^{T}A

^{T}, where

**Solution:**

Given,

(AB)

^{T}= B^{T}A^{T}⇒

⇒

⇒

⇒

⇒ L.H.S = R.H.s

So,

(AB)

^{T}= B^{T}A^{T}

**Question 7: Find ****, A**^{T} – B^{T}

^{T}– B

^{T}

**Solution:**

Given that

We need to find A

^{T}– B^{T}.Given that,

Let us find A

^{T}– B^{T}⇒

⇒

⇒

**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

**Question 10: If l**_{i}, m_{i}, n_{i} ; i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AA^{T} = I,

_{i}, m

_{i}, n

_{i}; i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AA

^{T}= I,

**Where **

**Solution:**

Given,

l

_{i}, m_{i}, n_{i}are direction cosines of three mutually perpendicular vectors⇒

And,

Given,

= I

Hence,

AA

^{T}= I