AB = 0,if and only if | vedclass

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Question

(D) The condition $AB = O$ for matrices does not necessarily imply that $A = O$ or $B = O$. In matrix algebra,there exist non-zero matrices $A$ and $B

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) The condition $AB = O$ for matrices does not necessarily imply that $A = O$ or $B = O$. In matrix algebra,there exist non-zero matrices $A$ and $B$ such that their product $AB$ is the zero matrix $O$. For example,if $A = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}$,then $AB = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = O$,even though $A 
eq O$ and $B 
eq O$. Therefore,none of the given options $A 
eq O, B = O$,$A = O, B 
eq O$,or $A = O$ or $B = O$ are necessary conditions for $AB = O$. Thus,the correct option is $(d)$.

$AB = 0$,if and only if

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