7.5 Matrices and matrix operations  (Page 4/10)

First, we check the dimensions of the matrices. Matrix $A$ has dimensions $2\times 2$ and matrix $B$ has dimensions $2\times 2.$ The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions $2\times 2.$

We perform the operations outlined previously.

1. As the dimensions of $A$ are $2\times 3$ and the dimensions of $B$ are $3\times 2,$ these matrices can be multiplied together because the number of columns in $A$ matches the number of rows in $B.$ The resulting product will be a $2\times 2$ matrix, the number of rows in $A$ by the number of columns in $B.$
   $$\begin{array}{l}\hfill \\ AB=\left[\begin{array}{rrr}\hfill \mathrm{-1}& \hfill 2& \hfill 3\\ \hfill 4& \hfill 0& \hfill 5\end{array}\right]\left[\begin{array}{rr}\hfill 5& \hfill \mathrm{-1}\\ \hfill -4& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right]\hfill \\ =\left[\begin{array}{rr}\hfill \mathrm{-1}(5)+2(\mathrm{-4})+3(2)& \hfill \mathrm{-1}(\mathrm{-1})+2(0)+3(3)\\ \hfill 4(5)+0(\mathrm{-4})+5(2)& \hfill 4(\mathrm{-1})+0(0)+5(3)\end{array}\right]\hfill \\ =\left[\begin{array}{rr}\hfill \mathrm{-7}& \hfill 10\\ \hfill 30& \hfill 11\end{array}\right]\hfill \end{array}$$
2. The dimensions of $B$ are $3\times 2$ and the dimensions of $A$ are $2\times 3.$ The inner dimensions match so the product is defined and will be a $3\times 3$ matrix.
   $$\begin{array}{l}\hfill \\ BA=\left[\begin{array}{rr}\hfill 5& \hfill \mathrm{-1}\\ \hfill \mathrm{-4}& \hfill 0\\ \hfill 2& \hfill 3\end{array}\right]\left[\begin{array}{rrr}\hfill \mathrm{-1}& \hfill 2& \hfill 3\\ \hfill 4& \hfill 0& \hfill 5\end{array}\right]\hfill \\ =\left[\begin{array}{rrr}\hfill 5(\mathrm{-1})+\mathrm{-1}(4)& \hfill 5(2)+\mathrm{-1}(0)& \hfill 5(3)+\mathrm{-1}(5)\\ \hfill \mathrm{-4}(\mathrm{-1})+0(4)& \hfill \mathrm{-4}(2)+0(0)& \hfill \mathrm{-4}(3)+0(5)\\ \hfill 2(\mathrm{-1})+3(4)& \hfill 2(2)+3(0)& \hfill 2(3)+3(5)\end{array}\right]\hfill \\ =\left[\begin{array}{rrr}\hfill \mathrm{-9}& \hfill 10& \hfill 10\\ \hfill 4& \hfill \mathrm{-8}& \hfill \mathrm{-12}\\ \hfill 10& \hfill 4& \hfill 21\end{array}\right]\hfill \end{array}$$