Multiplication of matrices
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Given \(X = \left[ {\begin{array}{*{20}{c}}2&1\\3&3\\1&2\end{array}} \right]\) and \(Y = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\), then \(XY=\)
\begin{align}
&X=\left[\begin{array}{ll}
2 & 1 \\
3 & 3 \\
1 & 2
\end{array}\right],\;\;\;\; Y=\left[\begin{array}{l}
2 \\
1
\end{array}\right] \\
&\qquad 3 \times 2 \rightarrow 2 \times 1 \rightarrow 3 \times 1 \\
&\begin{aligned}
XY&=\left[\begin{array}{ll}
2 & 1 \\
3 & 3 \\
1 & 2
\end{array}\right]\left[\begin{array}{l}
2 \\
1
\end{array}\right]\\
&=\left[\begin{array}{l}
2 \times 2+1 \times 1 \\
3 \times 2+3 \times 1 \\
1 \times 2+2 \times 1
\end{array}\right]\\
&=\left[\begin{array}{l}
5 \\
9 \\
4
\end{array}\right]
\end{aligned}
\end{align}
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