# Help with Matrix Problem. Find a matrix A that satisfies the equation AB = C.

Let $$B = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 1 \\ 1 & 3 & 5 \end{bmatrix}$$ and $$C = \begin{bmatrix} 1 & 3 & 5 \\ 1 & 2 & 3 \\ 1 & 1 & 1 \end{bmatrix}.$$

Find a matrix $A$ that satisfies the equation AB = C. I tried to do A = C*B^(-1) but found the determinant is 0 for B. Is there another way to solve this?

• sure. Multiplying on the left by an "elementary"matrix $A_1$ causes a row operation on $B.$ Multiplying on the left by $A_2$ causes a second row operation. Instead of trying to get reduced form, you try to get $C.$ The result is $A = A_k A_{k-1} \cdots A_2 A_1$ works – Will Jagy Feb 7 '18 at 2:20
• Have you noticed that $C$ is just $B$ with the rows permuted? Do you know how to find a matrix such that multiplying by it permutes the rows? – Gerry Myerson Feb 8 '18 at 6:06

Use row-operations on $B^\top A^\top = C^\top$.
Since $B$ is singular, there is no guarantee that the equation has a solution (it will depend on the columns of $C^\top$).