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 $\displaystyle C=AB$

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

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

$\displaystyle =AIA^{-1}$

$\displaystyle =AA^{-1}$

$\displaystyle =I$

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

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

$\displaystyle =I$

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

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

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

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

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

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

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

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

$\displaystyle \Rightarrow \det(AB)\ne0$

$\displaystyle \Rightarrow AB$ is non-singular