# How To Find Matrix B Given Matrix AB and A?

Matrix A:

\begin{bmatrix} 1 & -2 \\ -2 & 5 \end{bmatrix}

Product of Matrices AB:

\begin{bmatrix} 1 & 2 & -1 \\ 6 & -9 & 3\end{bmatrix}

Find Matrix B?

I am assuming that matrix B is 2x3 matrix but how does one go about finding it ?

• Multiply $A^{-1}$ by $AB$. – Filippo De Bortoli Sep 28 '14 at 11:06
• $A^{-1} A B = B$ – user137794 Sep 28 '14 at 11:06

(1) Write down your matrix $B$ as a matrix of unknowns. Then do the matrix multiplication of $A$ and $B$ and equate it to the known $AB$, giving you a system of equations to solve.
(2) If matrix $A$ is invertible, use its inverse.
We have that $A^{-1}AB=B$ where $A^{-1}$ is the inverse of the (square) matrix $A$