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 A^{4}
Solution :
A^{4} = A ⋅ A ⋅ A ⋅ A
= A^{2 }⋅ A^{2}
In order to find A^{4}, let us multiply A^{2} and A^{2}
Question 2 :
Consider the matrix
(i) Show that A_{α} A_{β} = A_{(α+β)}
(ii) Find all possible real values of α satisfying the condition A_{α + }A_{α}^{T} = I
Solution :
From the given question, if
By multiplying A_{α} A_{β}, 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 A^{2} is a unit matrix.
Solution :
Question 5 :
If A =
and A^{3} - 6A^{2} + 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 A^{2} = A, find the value of 7A - (I + A)^{3}.
Solution :
= 7A - (I + A)^{3}
= 7A - (I^{3} + 3A^{2}I + 3AI^{2} + A^{3})
= 7A - (I + 3AI + 3AI + A^{2} ⋅ A)
The product of identity matrix and A is matrix A.
= 7A - (I + 3A + 3A + A ⋅ A)
= 7A - (I + 6A + A^{2})
= 7A - (I + 6A + A)
= 7A - (I + 7A)
= 7A - I - 7A
= - I
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