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If A is nonsingular and B is nonsingular show that A - B is nonsingular

is this true or false and why ...

my answer

False cuase

A is nonsingular then det A =/ 0

B is nonsingular then det B =/ 0

det ( A - B ) =/ 0

det A - det B =/ 0

det A =/ det B

that mean that A - B nonsingular if and only if det A =/ det B

is my answer right

thankful

Sadeem Zone
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    $\det(A-B)$ is not $\det A-\det B$, and it appears to be all there is to say of your work. –  Apr 27 '17 at 17:32

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@ Sadeem Zone , indeed, you still have some way to go. But don't worry, you are almost right.

If the entries of $A,B$ follow (independently) the normal law $N(0,1)$, then $prob(A,B,A+B \text{ are invertible })=1$.

More generally, if $P,Q,R,S$ are randomly chosen (as above), then $Pdiag(I_p,0_{n-p})Q+Rdiag(0_p,I_{n-p})S$ is invertible with probability $1$.