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# Find the products $AB$ and $BA$ for the diagonal matrices.$A=$$\begin{bmatrix} 3&0&0\\ 0&-5&0\\ 0&0&0 \end{bmatrix}$$B=$$\begin{bmatrix} -7&0 &0\\ 0&4 &0 \\ 0&0 &12 \end{bmatrix}$

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Before carrying out any product between matrices $(AB)$, it is vital to check that it is defined.

Remember that for the product of two matrices to be defined, the column number of the first matrix (the matrix that is furthest to the left) has to be equal to the row number of the second matrix. If $A$ is an $m\times k$ matrix and $B$ is $k \times n$, then the product of $AB$ gives us an $m\times n$ matrix $C$.

Note that each element $c_{ij}$ of the resulting matrix $C$ is the sum of the products of each element of a row $i$ of $A$ with its corresponding element of column $j$ of $B$. This is

$\textcolor{#4257B2}{c_{ij} =\sum_{p=1}^{k} a_{i p} b_{p j}}\tag{1}$

or

$c_{ij}=a_{i 1} b_{1 j}+a_{i 2} b_{2 j}+a_{i 3} b_{3 j}+\cdots+a_{i k} b_{k j}.$

Is the matrix product commutative?

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