# A is a 3x3 matrix and A^-1 = ([3, 0, -1],[0, 8, 7],[-2, 3, 4]). If B is another matrix and BA = ([4, -3, 7],[-1, 0, 2]), how do you find the matrix B?

Apr 4, 2016

$B = \left(\begin{matrix}- 2 & - 3 & 3 \\ - 7 & 6 & 9\end{matrix}\right)$

#### Explanation:

Note that $A {A}^{-} 1 = I$ where $I$ is the $3 \text{x} 3$ identity matrix. As matrix multiplication is associative, we have

$\left(B A\right) {A}^{- 1} = B \left(A {A}^{- 1}\right) = B I = B$

Thus, to find $B$, we can simply multiply $B A$ by ${A}^{- 1}$. Doing so, we have:

$B = B A {A}^{- 1}$

$= \left(\begin{matrix}4 & - 3 & 7 \\ - 1 & 0 & 2\end{matrix}\right) \left(\begin{matrix}3 & 0 & - 1 \\ 0 & 8 & 7 \\ - 2 & 3 & 4\end{matrix}\right)$

$= \left(\begin{matrix}- 2 & - 3 & 3 \\ - 7 & 6 & 9\end{matrix}\right)$