# How can I divide two matrices together?

## For example, let A and B be 3x3 Matrices = What is A/B? How can I compute that?

Feb 12, 2017

By computing the "divisor" matrix's inverse, then post-multiplying the "dividend" matrix by this inverse.

#### Explanation:

Strictly speaking, division of matrices is not possible. But we can get around that by remembering that division can also be thought of as "multiplication by an inverse".

For instance, if we wanted to "divide" $A \div B$, we'd first compute the inverse ${B}^{-} 1$, and then multiply $A \times {B}^{-} 1$. Note that this means $B$ must be invertible, and so we need it to be a square matrix.

You can use Gauss-Jordan elimination on the matrix $B$ augmented with the identity matrix ${I}_{n}$ on the right to find the inverse of $B$:

$\left[\begin{matrix}\null & \null & \null & | & 1 & 0 & 0 \\ \null & B & \null & | & 0 & 1 & 0 \\ \null & \null & \null & | & 0 & 0 & 1\end{matrix}\right]$

Row-reducing $B$ into $I$ on the left turns $I$ into ${B}^{-} 1$ on the right:

$\implies \left[\begin{matrix}1 & 0 & 0 & | & \null & \null & \null \\ 0 & 1 & 0 & | & \null & {B}^{-} 1 & \null \\ 0 & 0 & 1 & | & \null & \null & \null\end{matrix}\right]$

There are other ways, but this is one of the more common.

Once you have ${B}^{-} 1$, post-multiply $A$ by ${B}^{-} 1$ to get your "quotient".

Hope this helps!