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Question Number 199001 by mnjuly1970 last updated on 26/Oct/23

$$$$$$\mathrm{I}f,\mathrm{A}\in {\mathrm{M}}_{n\times n},{\mathrm{A}}^2=\mathrm{A},1\ne k\in \mathbb{R}.$$$$\mathrm{F}ind{(\mathrm{I}-k\mathrm{A})}^{-1}=?$$

Answered by MM42 last updated on 26/Oct/23

$$if\mid A\mid \ne 0$$$$\Rightarrow A^2=A\stackrel{\times A^{-1}}{\Rightarrow }A=I$$$$\Rightarrow {(I-kA)}^{-1}={(I-kI)}^{-1}=\frac{1}{1-k}I✓$$$$$$

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