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# Let $A\;=\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}, B\;=\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix}, C\;=\begin{bmatrix}-2 & 5\\3 & 4\end{bmatrix}$. Find $AB\qquad$

$\begin{array}{1 1} -19 \\ 19 \\ 18 \\ -18 \end{array}$

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Toolbox:
• Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix.
• 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:
• $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
Given $A\;=\begin{bmatrix}2 & 4\\3 & 2\end{bmatrix}$ and $B\;=\begin{bmatrix}1 & 3\\-2 & 5\end{bmatrix}$,
$AB_{1,1}$ = 2x1 + 4x(-2) = -6
$AB_{1,2}$ = 2x3+4x5 = 26
$AB_{2,1}$ = 3x1 + 2x(-2) = -1
$AB_{2,2}$ = 3x3 + 2x5 = 19
$\begin{bmatrix}AB\end{bmatrix} = \begin{bmatrix}-6&26 \\ -1&19\end{bmatrix}$.
answered Feb 27, 2013