Question:
If $I$ is the identity matrix and $A$ is a square matrix such that $A^{2}=A$, then what is the value of $(I+A)^{2}=3 A$ ?
Solution:
Given: A is a square matrix, such that $A^{2}=A$.
Here,
$(I+A)^{2}-3 A=(I+A)(I+A)-3 A$
$\Rightarrow(I+A)^{2}-3 A=I \times I+I \times A+A \times I+A \times A-3 A \quad$ (using distributive property)
$\Rightarrow(I+A)^{2}-3 A=I+A+A+A^{2}-3 A \quad($ using $I \times I=I$ and $I A=A I=A)$
$\Rightarrow(I+A)^{2}-3 A=I+2 A+A-3 A \quad\left(\because A^{2}=A\right)$
$\Rightarrow(I+A)^{2}-3 A=I+3 A-3 A$
$\Rightarrow(I+A)^{2}-3 A=I$
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