If A and B are two non-zero n $\times$ n matrices such that $\mathrm{A^2+B=A^2B}$, then :
Solution
Given : $A^{2}+B=A^{2} B\quad...(i)$
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$\Rightarrow A^{2}+B-I=A^{2} B-I$
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$\Rightarrow A^{2} B-A^{2}-B+I=I$
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$\Rightarrow A^{2}(B-I)-I(B-I)=I$
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$\Rightarrow\left(A^{2}-I\right)(B-I)=I$
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$\therefore A^{2}-I$ is the inverse matrix of $B-I$ and vice versa.
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So, $(B-I)\left(A^{2}-I\right)=I$
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$\Rightarrow B A^{2}-B-A^{2}+I=I$
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$\therefore A^{2}+B=B A^{2} \quad...(ii)$
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So, by (i) and (ii)
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$A^{2} B=B A^{2}$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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