Answer
$AB=\left[\begin{array}{rrr}{-2}&{1}&{-2}\\{10}&{-2}&{2}\\{-10}&{2}&{-2}\end{array}\right]$
Work Step by Step
If $A$ is an $m\times\boxed{n }$ matrix and $B$ is an $\boxed{n }\times k$ matrix,
then the product $AB$ is the $m\times k$ matrix whose $ij-$th entry is the product
of the (ith row in A) and (j-th column in B)
$(AB)_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+a_{i3}b_{3j}+\cdots+a_{in}b_{nj}$.
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A is a 3$\times$2 matrix, B is a 2$\times$3 matrix.
AB is defined and is a 3$\times$3 matrix
$AB=\left[\begin{array}{rr}{1}&{-1}\\{0}&{2}\\{0}&{-2}\end{array}\right]\left[\begin{array}{rrr}{3}&{0}&{-1}\\{5}&{-1}&{1}\end{array}\right]$
$=\left[\begin{array}{rrr}{1 \cdot 3+(-1) \cdot 5}&{1\cdot 0+(-1)(-1)}&{1\cdot(-1)+(-1)\cdot 1}\\{0\cdot 3+2\cdot 5}&{0\cdot 0+2(-1)}&{0\cdot(-1)+2\cdot 1}\\{0\cdot 3+(-2)\cdot 5}&{0\cdot 0+(-2)(-1)}&{0\cdot(-1)+(-2)\cdot 1}\end{array}\right]$
$=\left[\begin{array}{rrr}{-2}&{1}&{-2}\\{10}&{-2}&{2}\\{-10}&{2}&{-2}\end{array}\right]$
