4 Multiplying two 3 $\times$ 3 matrices

The definition of the product $C=AB$ where $A$ and $B$ are two $3\times 3$ matrices is as follows

$C=\left[\begin{array}{ccc}\hfill a\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill d\hfill & \hfill e\hfill & \hfill f\hfill \\ \hfill g\hfill & \hfill h\hfill & \hfill i\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill r\hfill & \hfill s\hfill & \hfill t\hfill \\ \hfill u\hfill & \hfill v\hfill & \hfill w\hfill \\ \hfill x\hfill & \hfill y\hfill & \hfill z\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill ar+bu+cx\hfill & \hfill as+bv+cy\hfill & \hfill at+bw+cz\hfill \\ \hfill dr+eu+fx\hfill & \hfill ds+ev+fy\hfill & \hfill dt+ew+fz\hfill \\ \hfill gr+hu+ix\hfill & \hfill gs+hv+iy\hfill & \hfill gt+hw+iz\hfill \end{array}\right]$

This looks a rather daunting amount of algebra but in fact the construction of the matrix on the right-hand side is straightforward if we follow the simple rule from Key Point 4 that the element in the $i$ ^th row and $j$ ^th column of $C$ is obtained by multiplying the $i$ ^th row of $A$ with the $j$ ^th column of $B$ .

For example, to obtain the element in row 2, column 3 of $C$ we take row 2 of $A$ : $\left[d,e,f\right]$ and multiply it with column 3 of $B$ in the usual way to produce $\left[dt+ew+fz\right]$ .

By repeating this process we obtain every element of $C$ .

Task!

Calculate $AB=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill -1\\ \hfill 3& \hfill 4& \hfill 0\\ \hfill 1& \hfill 5& \hfill -2\end{array}\right]\left[\begin{array}{ccc}\hfill 2& \hfill -1& \hfill 3\\ \hfill 1& \hfill -2& \hfill 1\\ \hfill 0& \hfill 3& \hfill -2\end{array}\right]$

First find the element in row 2 column 1 of the product:

Answer

Now complete the multiplication to find all the elements of the matrix $AB$ :

Answer

The $3\times 3$ unit matrix is: $I=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$   andas in the $2\times 2$ case this has the property that    $AI=IA=A$

The $3\times 3$ zero matrix is $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$