Answer
$P=\left[\begin{array}{rrr}
5 & 2 & 1\\
7 & 10 & -7
\end{array}\right]$
Work Step by Step
If $A$ is an $m\times n$ matrix and $B$ is an $n\times k$ matrix
(so the number of columns of $A$ is the same as the number of rows of $B$),
then the matrix product $AB$ is the $m\times k$ matrix
whose $ij$-entry is the inner product of the $i\mathrm{t}\mathrm{h}$ row of $A$ and the jth column of $B.$
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The first matrix is a 2$\times\color{blue}{2}$ matrix, and the second is a $\color{blue}{2} \times 3$ matrix.
So, the product is defined, and it is a 2$\times 3$ matrix.
Naming the product matrix $P=[p_{ij}],$
$p_{11}$ = (row 1 in A) times (column 1 in B)$=1(1)+2(2)=5$
$p_{12}$ = (row 1 in A) times (column $2$ in B)$=1(-2)+2(2)=2$
$p_{13}$ = (row 1 in A) times (column $3$ in B)$=1(3)+2(-1)=1$
$p_{21}$ = (row $2$ in A) times (column 1 in B)$=-1(1)+4(2)=7$
$p_{22}$ = (row $2$ in A) times (column $2$ in B)$=-1(-2)+4(2)=10$
$p_{22}$ = (row $2$ in A) times (column $3$ in B)$=-1(3)+4(-1)=-7$
$P=\left[\begin{array}{lll}
5 & 2 & 1\\
7 & 10 & -7
\end{array}\right]$
