If A is square matrix such that A^2 = A, then (I + A)^3 – 7 A is … - Vidyadip

Question

If A is square matrix such that $A^2 = A,$ then $(I + A)^3 – 7\ A$ is equal to

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Answer

Given that $A^2 = A$
Calculating value of $(I + A)^3 - 7A:$
$(I+A)^3 - 7A=I^{3}+A^{3}+3 I^{2} A+3 I A^{2}-7 A$
$= \mathrm{I}+\mathrm{A}^{2} \cdot \mathrm{A}+3 \mathrm{A}+3 \mathrm{A}^{2}-7 \mathrm{A}\left(\mathrm{I}^{\mathrm{n}}=\mathrm{I} \text { and } \mathrm{I} \cdot \mathrm{A}=\mathrm{A}\right)$
$= I+A \cdot A+3 A+3 A-7 A\left(A^{2}=A\right)$
$= I+A^{2}+3 A+3 A-7 A$
$= I +7A-7A$
Hence,$ (I + A)^3 - 7A = I$

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