# Math Help - Matrix Probelm

1. ## Matrix Probelm

Let A and B be nonsingular matrices of the same order. Show that AB is nonsigular and (AB)^-1 = B^-1 * A^-1

Hint: Multiply by AB

Any advice...I don't know how to show this

Thanks.

2. Originally Posted by jzellt
Let A and B be nonsingular matrices of the same order. Show that AB is nonsigular and (AB)^-1 = B^-1 * A^-1

Hint: Multiply by AB

Any advice...I don't know how to show this

Thanks.
HI

Let $C=AB$

$C(B^{-1}A^{-1})=AB(B^{-1}A^{-1})$

$=A(BB^{-1})A^{-1}$

$=AIA^{-1}$

$=AA^{-1}$

$=I$

Similarly , $(B^{-1}A^{-1})C=(B^{-1}A^{-1})(AB)$

$=B^{-1}(AA^{-1})B$

$=I$

Since $(B^{-1}A^{-1})C=C(B^{-1}A^{-1})$

For both cases , $C^{-1}=B^{-1}A^{-1}$

$
(AB)^{-1}=B^{-1}A^{-1}
$

3. ## Determinant of Matrix Product

Hello jzellt
Originally Posted by jzellt
Let A and B be nonsingular matrices of the same order. Show that AB is nonsigular and (AB)^-1 = B^-1 * A^-1

Hint: Multiply by AB

Any advice...I don't know how to show this

Thanks.
mathaddict has given you the proof that $(AB)^{-1}=B^{-1}A^{-1}$.

The first part of the proof (that $AB$ is non-singular) is very straightforward:
$A$ and $B$ are non-singular

$\Rightarrow\det(A)\ne0$ and $\det(B)\ne0$

$\Rightarrow \det(A)\det(B) \ne 0$

But $\det(A)\det(B)=\det(AB)$

$\Rightarrow \det(AB)\ne0$

$\Rightarrow AB$ is non-singular