# Operations on 3x3 matrix through matrix products

What would I have to multiply the following matrix ... $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}$$

by so that I get the following matrices below?

1. $$\begin{bmatrix} d & e & f \\ a & b & c \\ g & h & i \\ \end{bmatrix}$$

2.

$$\begin{bmatrix} a & b & c \\ g & h & i \\ d & e & f \\ \end{bmatrix}$$

3.

$$\begin{bmatrix} g & h & i \\ d & e & f \\ a & b & c \\ \end{bmatrix}$$

• Hint: right-multiplying by the $j$th column of the identity matrix selects the $j$th column of the matrix; left-multiplying by the $i$th row of the identity selects the $i$th row. – amd Feb 6 '16 at 1:53
• Is the upper-right corner element of #3 supposed to be $i$? – amd Feb 6 '16 at 1:54
• @amd yes, the upper-right corner element of no.3 is supposed to be "i" – Dan Feb 6 '16 at 1:56
• @amd Is there a way to backsolve matrix multiplication so that I can use #3 and the matrix $$\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix}$$ to derive which matrix I should multiply matrix in #3 by to get matrix#3? – Dan Feb 6 '16 at 1:58
• @Tian_He Matrix products are in the form Ax=B. When I know what A and B are, what's the quickest way to find x? – Dan Feb 6 '16 at 2:04

## 2 Answers

Left multiply the original matrix by the following rearranged identity matrix, and see what happens. You should be able to get an idea of the purpose of this exercise. \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix}

Every one of those matrices is just the original with its rows permuted. As I mentioned in my comment, left-multiplying a matrix by the $i$th row of the identity picks out its $i$th row. For example, $$\left[\begin{matrix}0&1&0\end{matrix}\right] \left[\begin{matrix}a&b&c\\d&e&f\\g&h&i\end{matrix}\right] = \left[\begin{matrix}d&e&f\end{matrix}\right].$$ So, for the first problem, the answer is going to be of the form $$\left[\begin{matrix}0&1&0\\ \cdots&\cdots&\cdots \\ \cdots&\cdots&\cdots \end{matrix}\right].$$ Can you fill in the rest of it now?