If A is a square matrix such that A^2=A, then (1+A)^3-7 A is equa…

MCQ

If $A$ is a square matrix such that $A^2=A$, then $(1+A)^3-7 A$ is equal to:

A
A
B
I - A
✓
I
D
3A

✓

Answer

Correct option: C.

Given: $A^2=A \ldots (i)$
Multiplying both sides by $A, A^3=A^2=A[$ From eq. (i) $] \ldots (ii)$
Also given $(I+A)^3-7 A=I^3+A^3+3 I^2 A+3 I A^2-7 A$
Putting $A^2=A\left[\right.$ from eq. (i)] and $A^3=A[$ from eq. $(ii)],$
$=I+A+3 I A+3 I A-7 A=I+A+3 A+3 A-7 A[\because I A=A]$
$=I+7 A-7 A=I$
Therefore, option $(C)$ is correct.

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