If A is a square matrix such that A^2=A, then (I-A)^3+A is equal … - Vidyadip

MCQ

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

A
I
B
O
C
I-O
D
l+A

✓

Answer

We have, $A^2=A$
$
\begin{array}{l}
\text { Now, }(I-A)^3+A=(I-A)(I-A)(I-A)+A \\
=(I \cdot I-I \cdot A-A \cdot I+A \cdot A)(I-A)+A \\
=(I-A-A+A)(I-A)+A \quad\left[\because I \cdot A=A \cdot I=A \text { and } A^2=A\right] \\
=(I-A)(I-A)+A=(I \cdot I-I \cdot A-A \cdot I+A \cdot A)+A \\
=(I-A-A+A)+A=(I-A)+A=I
\end{array}
$

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