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CBSE Class 12 · 2016 · MatricesVery Short Answer

If

A

is a square matrix such that

A^{2} = I

, then find the simplified value of

(A - I)^{3} + (A + I)^{3} - 7 A

.

Answer: $A$

Solution

Expand both cubes (as

A

commutes with

I

):

(A - I)^{3} + (A + I)^{3} = 2 A^{3} + 6 A I^{2} = 2 A^{3} + 6 A .

Since

A^{2} = I

,

A^{3} = A \cdot A^{2} = A

, so

2 A^{3} + 6 A = 2 A + 6 A = 8 A

. Therefore

(A - I)^{3} + (A + I)^{3} - 7 A = 8 A - 7 A = A .

A

CBSE Class 12 Board · 2016 · Subjective

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