Engineering
Mathematics
Introduction to Matrix
Algebra of Matrices
Special Type of Square Matrices
Question
If A is a square matrix such that A2 = I, then find the simplified value of (A – I)3 + (A + I)3 – 7A.
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Solution
A2 = I
∴ A3 = A2A = IA = A
We know
(A + B)3 = A3 + 3A2B + 3AB2 + B3
(A – B)3 = A3 – 3A2B + 3AB2 – B3
Provided that AB = BA
Since, AI = IA = A
∴ (A + I)3 = A3 + 3A2I + 3AI2 + I3
and, (A – I)3 = A3 – 3A2I + 3AI2 – I3
⇒ (A + I)3 = A3 + 3A2 + 3A + I
and , (A – I)3 = A3 – 3A2 + 3A – I
∴ (A + I)3 + (A – I)3 = 2(A3 + 3A)
= 2(A + 3A)
Hence, = 84
(A – I)3 + (A + I)3 – 7A = 8A – 7A = A