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# If $\begin{bmatrix}2 & 1 & 3\end{bmatrix}\begin{bmatrix}-1 & 0 & -1\\-1 & 1 & 0\\0 & 1 & 1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=A,find\;A.$

Toolbox:
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Step1:
Let $B=\begin{bmatrix}2 & 1 & 3\end{bmatrix}$
$C=\begin{bmatrix}-1 & 0 & -1\\-1 & 1 & 0\\0 & 1 & 1\end{bmatrix}$
$D=\begin{bmatrix}1\\0\\-1\end{bmatrix}$
$BC=\begin{bmatrix}2 & 1 & 3\end{bmatrix}\begin{bmatrix}-1 & 0 & -1\\-1 & 1 & 0\\0 & 1 & 1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}2(-1)+1(-1)+3(0)&2(0)+1(1)+3(1)&2(-1)+1(0)+3(1)\end{bmatrix}$
$\Rightarrow \begin{bmatrix}-2-1+0&0+1+3&-2+0+3\end{bmatrix}$
$\Rightarrow \begin{bmatrix}-3&4&1\end{bmatrix}$
Step2:
Now (BC)D=A
$A= \begin{bmatrix}-3&4&1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}$
$=\begin{bmatrix}-3(1)+4(0)+1(-1)\end{bmatrix}$
$=\begin{bmatrix}-3+0-1\end{bmatrix}$
$=\begin{bmatrix}-4\end{bmatrix}$
$\Rightarrow A=[-4].$