#### 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.

#### Solution

Given:

(A−I)^{3}+(A+I)^{3}−7A

=A^{3}−I^{3}−3A^{2}I+3AI^{2}+A^{3}+I^{3}+3A^{2}I+3AI^{2}−7A

= 2A^{3}+6AI^{2}−7A

=2A.A^{2}+6AI^{2}−7A

=8A−7A

=A

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Solution If A is a square matrix such that A2 = I, then find the simplified value of (A – I)3 + (A + I)3 – 7A. Concept: Types of Matrices.