Which of the following is incorrect?

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(A) For option $(a)$,we expand the product of matrices: $(A + B)(A - B) = A^2 - AB + BA - B^2$. Since matrix multiplication is generally not commutati

Vedclass · Accepted Answer

$A^2 - B^2 = (A + B)(A - B)$

Vedclass · Answer

(A) For option $(a)$,we expand the product of matrices: $(A + B)(A - B) = A^2 - AB + BA - B^2$. Since matrix multiplication is generally not commutative $(AB 
eq BA)$,we cannot simplify this to $A^2 - B^2$ unless $A$ and $B$ commute. Therefore,the statement $A^2 - B^2 = (A + B)(A - B)$ is generally incorrect. For option $(b)$,the transpose of a transpose is the original matrix,which is a standard property. For option $(c)$,if $A$ and $B$ commute,then $(AB)^n = A^n B^n$ is a valid property. For option $(d)$,$(A - I)(A + I) = A^2 + AI - IA - I^2 = A^2 - I$. Setting this to $O$ gives $A^2 - I = O$,or $A^2 = I$. Thus,option $(a)$ is the incorrect statement.

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