Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
add comment

2 Answers 2

up vote 5 down vote accepted

Hint: Compute the determinant of $AB$.

share|improve this answer
add comment

$A$ 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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.