Question

If $A$ is a square matrix such that $A^2=I$, then $(A-I)^3+(A+I)^3-7 A$ is equal to [NCERT Exemplar]

• A. $A$
• B. $l-A$
• C. $1+A$
• D. $3 A$

Answer 1

Answer: (A)

Given that, $A^2=1$
Consider, $(A-I)^3+(A+I)^3-7 A$
$=A^3-I^3-3 A^2 l+3 A I^2+A^3+I^3+3 A^2 I+3 A I^2-7 A$
$\begin{bmatrix}\because(a+b)^3=a^3+b^3+3 a^2 b+3 a b^2 \\ \text { and }(a-b)^3=a^3-b^3-3 a^2 b+3 a b^2\end{bmatrix}$
$=2 A^3+6 A I^2-7 A$
$=2 A^2 \cdot A+6 A I-7 A \left(\because A^3=A^2 \cdot A\right.$ and $\left.I^2=I\right)$
$=2 l \cdot A+6 A-7 A$$\left(\because A^2=l\right)$
$=2 A+6 A-7 A$
$=8 A-7 A$
$=A$