Choose the correct answer from the given four options. If A is a … - Vidyadip

MCQ

Choose the correct answer from the given four options. If $A $ is a square matrix such that $A^2 = I,$ then $(A - I)^3 + (A + I)^3 - 7A$ is equal to:

✓
$A$
B
$I - A$
C
$I + A$
D
$3A$

✓

Answer

Correct option: A.

$A$

We have, $A^2 = I$
Now, $(A - I)^3 + (A + I)^3 - 7A $
$= [(A - I) + (A + I)][(A - I)^2 + (A + I)^2 - (A - I)(A + I)] - 7A$
$[\because$ $a^3 + b^3 = (a + b)(a^2 + b^2 - ab)]$
$= [(2A){A^2 + I^2 - 2AI + A^2 + I^2 + 2AI - (A^2 - I^2)}] - 7A$
$= [(2A){AI + I^2 - 2AI + AI + I^2 + 2AI - AI +I^2}] - 7A [\because A^2 = AI ]$
$= 2A[I + I^2 + I + I^2 - I + I^2] - 7A$
$= 2A[5I - I] - 7A$
$= 8AI - 7AI [\because A = AI ]$
$= AI$
$= A$

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