Question

If A is a square matrix such that A^2=I,then find the simplified value of (A-I)^3+(A+I)^3-7A.

Answer 1

ANSWER:-
Given:-A^2=I
(A−I)3+(A+I)3−7A.

We know that
(a+b)3=a3+b3+3ab(a+b)
(a−b)3=a3+b3−3ab(a+b)
(A−I)3+(A+I)3−7A.
A3−I3−3AI(A−I)+A3+I3+3AI(A+I)−7A.

I.A−I3−3A2I+3AI2+IA+I3+3A2I+3AI2−7A.

A−I3−3I3+3AI2+IA+I3+3I3+3AI2−7A
A+3A+A+3A−7A
8A−7A=A.
Therefore the simplified value is A.

Answer 2

Given : $A^2 = I$

and we have to find the value of:

$(A-I)^3 + (A+I)^3 - 7A$

Now, $(A+I)^3 = A^3+I^3+3AI(A+I)\\(A-I)^3 = A^3-I^3 - 3AI(A+I)$

Also given that , $A^2=I$

and we know that $I^n = I$

Hence the given equation becomes,

$2A^3 - 7A \\\\= 2A - 7A = 5A$

which is the final answer.