Here we are going to see how to multiply two matrices. To understand the concept, let us look into some example problems.

 Question 1 :If A  =

then compute A4

Solution :

A4  =  A ⋅ A ⋅ A ⋅ A

=  A⋅ A2

In order to find A4, let us multiply A2 and A2

Question 2 :

Consider the matrix

(i)  Show that Aα Aβ =  A(α+β)

(ii) Find all possible real values of α satisfying the condition Aα + AαT  =  I

Solution :

From the given question, if

By multiplying Aα Aβ, we get

In the given question, by replacing α by (α + β), we get

 Question 3 :If A  =

and such that (A− 2I)(A− 3I) = O, find the value of x.

Solution :

(A− 2I)(A− 3I) = O

By equating the corresponding terms, we get

2x - 2  = 0

x  =  2/2  =  1

Hence the value of x is 1.

Question 4 :

If A =

show that A2 is a unit matrix.

Solution :

Question 5 :

If A =

and A3 - 6A2 + 7A + KI  =  0, then find the value of k.

Solution :

21 - 30 + 7 + K  =  0

28 - 30 + k  =  0

-2 + k  =  0

k  =  2

The value of k is 2.

Question 6 :

Give your own examples of matrices satisfying the following conditions in each case:

(i) A and B such that AB  BA .

(ii) A and B such that AB = O = BA, A  O and B  O.

(iii) A and B such that AB = O and BA  O.

Question 7 :

Show that f (x) f ( y) = f (x + y), where f(x)  =

Solution :

Let us find the product of f(x) and f(y)

=  f(x + y)

Hence proved.

Question 8 :

If A is a square matrix such that A2 = A, find the value of 7A - (I + A)3.

Solution :

=  7A - (I + A)3

=  7A - (I3 + 3A2I + 3AI2 + A3)

=  7A - (I + 3AI + 3AI + A2 ⋅ A)

The product of identity matrix and A is matrix A.

=  7A - (I + 3A + 3A + A ⋅ A)

=  7A - (I + 6A + A2)

=  7A - (I + 6A + A)

=  7A - (I + 7A)

=  7A - I - 7A

=  - I

After having gone through the stuff given above, we hope that the students would have understood "Matrix Multiplication Worksheet Answers".

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