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

Question

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

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Answer

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

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