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If $A$ and $B$ are square matrices of order $n$ such that $A^{2}-B^{2}=(A-B)(A+B)$, then which of the following will be true?
Options:
Solution:
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Verified Answer
The correct answer is:
$A B=B A$
Given,
$\begin{aligned} A^{2}-B^{2} &=(A-B)(A+B) \\ &=A^{2}-B A+A B-B^{2} \end{aligned}$
$\begin{aligned} \Rightarrow & 0 &=-B A+A B \\ \Rightarrow & A B &=B A \end{aligned}$
Also, options (a), (b) and (d) satisfy the given condition none of those is a necessary condition for $A^{2}-B^{2}=(A-B)(A+B)$
$\begin{aligned} A^{2}-B^{2} &=(A-B)(A+B) \\ &=A^{2}-B A+A B-B^{2} \end{aligned}$
$\begin{aligned} \Rightarrow & 0 &=-B A+A B \\ \Rightarrow & A B &=B A \end{aligned}$
Also, options (a), (b) and (d) satisfy the given condition none of those is a necessary condition for $A^{2}-B^{2}=(A-B)(A+B)$
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