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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.

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

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  • $\begingroup$ 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. $\endgroup$ – amd Feb 6 '16 at 1:53
  • $\begingroup$ Is the upper-right corner element of #3 supposed to be $i$? $\endgroup$ – amd Feb 6 '16 at 1:54
  • $\begingroup$ @amd yes, the upper-right corner element of no.3 is supposed to be "i" $\endgroup$ – Dan Feb 6 '16 at 1:56
  • $\begingroup$ @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? $\endgroup$ – Dan Feb 6 '16 at 1:58
  • $\begingroup$ @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? $\endgroup$ – Dan Feb 6 '16 at 2:04
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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}

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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?

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