HOW TO FIND THE PRODUCT OF TWO MATRICES

Question 1 :

If 

find AB, BA and check if AB = BA? 

Solution :

AB  =  

BA  =  

AB and BA are not equal.

Question 2 :

Given that

verify that A(B + C) = AB + AC .

R.H.S :

AB  =

Now, let us add AB and AC

 Hence proved.

Question 3 :

Show that the matrices

satisfy commutative property AB = BA

Solution :

Hence the commutative property is true for the above matrices.

Question 4 :

Show that (i) A(BC) = (AB)C

(ii) (A−B)C = AC −BC (iii) (A−B)^T = A^T −B^T

Solution :

(i) A(BC) = (AB)C

First let us find the product of B and C.

Let us multiply the matrices A and BC.

Let us find the product of AB.

Multiplying the matrices AB and , we get

A(BC)  =  (AB)C

Hence it is proved.

(ii) (A−B)C = AC −BC 

Solution :

L.H.S

(A−B) C  =  

R.H.S :

AC  =  

BC  =  

(A−B)C = AC −BC 

(iii) (A−B)^T = A^T −B^T

Hence proved.

Question 5 :

If

then show that A² +B² = I .

Solution :

A²   =  A x A

B²  =  B x B

By adding the corresponding terms, we get 

Hence proved.

Question 6 :

If A = 

prove that AA^T = I .

Solution :

Question 7 :

Verify that A² = I when

Solution :

  =  I  (hence proved)

Question 8 :

Solution :

Question 9 :

verify that (AB)^T = B^TA^T

Solution :

Question 10 :

show that A² - 5A + 7I₂  =  0

Solution :

Hence proved.

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