**How to Find the Product of Two Matrices ?**

Here we are going to see some example problems based on finding 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 A^{2} +B^{2} = I .

**Solution :**

**A ^{2} = A x A**

B^{2} = 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 A^{2} = I when

**Solution :**

= I (hence proved)

**Question 8 :**

**Solution :**

**Question 9 :**

verify that (AB)^{T} = B^{T}A^{T}

**Solution :**

**Question 10 :**

show that A^{2} - 5A + 7I_{2} = 0

**Solution :**

Hence proved.

After having gone through the stuff given above, we hope that the students would have understood, "How to Find the Product of Two Matrices".

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