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Suppose that there is singular (non-invertible) matrix $A$. If it gets multiplied by any square matrix $B$, would $AB$ be singular?

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up vote 6 down vote accepted

Hint: Compute the determinant of $AB$.

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$A$ is singular iff there is some $x\neq 0$ such that $Ax = 0$.

Now consider $AB$. If $B$ is singular, there is some $x'\neq 0$ such that $Bx' = 0$, hence $AB x'=0$ and so $AB$ is singular. If $B$ is invertible, then let $x' = B^{-1} x$ (where $Ax = 0$), then $ABx' = Ax = 0$. Hence $AB$ is singular.

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