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 $A^2=1$, then $(A-1)^3+(A+1)^3 - 7$ A is equal to :

✓
$A$
B
$I - A$
C
$I + A$
D
$3A$

✓

Answer

Correct option: A.

$A$

We have, $A ^2=1$
Now, $(A-I)^3+(A+I)^3-7 A=[(A-I)+(A+I)]$
$\left[(A-I)^2+(A+I)^2-(A-I)(A+I)\right]-7 A$
${\left[\because a^3+b^3=(a+b)\left(a^2+b^2-a b\right)\right]}$
$=\left[(2 A)\left\{A^2+I^2-2 A I+A^2+I^2+2 A I-\left(A^2-I^2\right)\right\}\right]-7 A$
$=\left[(2 A)\left\{A I+I^2-2 A I+A I+I^2+2 A I-A I+I^2\right\}\right]-7 A\left[\because A^2=A I\right]$
$=2 A\left[I+I^2+I+I^2-I+I^2\right]-7 A$
$=2 A[5 I-I]-7 A$
$=8 A I-7 A I\ [\because A=A I]$
$=A I$
$=A$

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