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Wednesday, August 13, 2014

Matrix Multiplication

Matrix Multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic

If A is an n × m matrix and B is an m × p matrix,
\mathbf{A}=\begin{pmatrix}
 A_{11} & A_{12} & \cdots & A_{1m} \\
 A_{21} & A_{22} & \cdots & A_{2m} \\
\vdots & \vdots & \ddots & \vdots \\
 A_{n1} & A_{n2} & \cdots & A_{nm} \\
\end{pmatrix},\quad\mathbf{B}=\begin{pmatrix}
 B_{11} & B_{12} & \cdots & B_{1p} \\
 B_{21} & B_{22} & \cdots & B_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
 B_{m1} & B_{m2} & \cdots & B_{mp} \\
\end{pmatrix}

the matrix product AB (denoted without multiplication signs or dots) is defined to be the n × p matrix

\mathbf{A}\mathbf{B} =\begin{pmatrix}
 \left(\mathbf{AB}\right)_{11} & \left(\mathbf{AB}\right)_{12} & \cdots & \left(\mathbf{AB}\right)_{1p} \\
 \left(\mathbf{AB}\right)_{21} & \left(\mathbf{AB}\right)_{22} & \cdots & \left(\mathbf{AB}\right)_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
 \left(\mathbf{AB}\right)_{n1} & \left(\mathbf{AB}\right)_{n2} & \cdots & \left(\mathbf{AB}\right)_{np} \\
\end{pmatrix}

If the column and rows are in the same size it is defined.



EXAMPLE:

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