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Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMatrices1 Mark
Question
If A is square matrix such that A2 = A, then (I + A)3 – 7 A is equal to
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Answer

Given that A2 = 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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