Answer:
[tex]\:c_{11}=-2,\:\:\:c_{12}=-2[/tex]
[tex]\:c_{21}=5,\:\:\:c_{22}=8[/tex]
Step-by-step explanation:
Given the matrices
[tex]A=\begin{pmatrix}1&-1\\ 2&1\end{pmatrix}[/tex]
[tex]B=\begin{pmatrix}1&2\\ \:3&4\end{pmatrix}[/tex]
Calculating AB:
[tex]\begin{pmatrix}1&-1\\ \:\:2&1\end{pmatrix}\times \:\begin{pmatrix}1&2\\ \:\:3&4\end{pmatrix}=\begin{pmatrix}c_{11}&c_{12}\\ \:\:\:c_{21}&c_{22}\end{pmatrix}[/tex]
Multiply the rows of the first matrix by the columns of the second matrix
[tex]=\begin{pmatrix}1\cdot \:1+\left(-1\right)\cdot \:3&1\cdot \:2+\left(-1\right)\cdot \:4\\ 2\cdot \:1+1\cdot \:3&2\cdot \:2+1\cdot \:4\end{pmatrix}[/tex]
[tex]=\begin{pmatrix}-2&-2\\ 5&8\end{pmatrix}[/tex]
Hence,
[tex]\begin{pmatrix}c_{11}&c_{12}\\ \:\:\:c_{21}&c_{22}\end{pmatrix}=\begin{pmatrix}-2&-2\\ \:5&8\end{pmatrix}[/tex]
Therefore,
[tex]\:c_{11}=-2,\:\:\:c_{12}=-2[/tex]
[tex]\:c_{21}=5,\:\:\:c_{22}=8[/tex]
