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 = AT −BT
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 = AT −BT
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 AAT = I .
Solution :
Question 7 :
Verify that A2 = I when
Solution :
= I (hence proved)
Question 8 :
Solution :
Question 9 :
verify that (AB)T = BTAT
Solution :
Question 10 :
show that A2 - 5A + 7I2 = 0
Solution :
Hence proved.
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