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Multiplying matrices

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Engineering Maths B, Matrices
August 8th, 2008 by Craig Rose
Permalink: http://www.craigsmaths.com/ea003-engineering-mathematics-b/multiplying-matrices/

Engineering Maths B, Matrices
August 8th, 2008 by Craig Rose
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Multiplying matrices is fairly simple but there is a process to learn and a couple of rules to remember.

Scalar Multiplication

A scalar is a value that has only one dimension such as a real number. Matrices two dimensions - rows and columns. When multiplying a matrix by a scalar multiply each element of the matrix by the scalar.

For the matrix:

$\mathit{A}=\begin{bmatrix}a & b & c\\d & e & f\\g & h & i\end{bmatrix}$ and the scalar $x$ :

$x\mathit{A}=x\begin{bmatrix}a & b & c\\d & e & f\\g & h & i\end{bmatrix}=\begin{bmatrix}xa & xb & xc\\xd & xe & xf\\xg & xh & xi\end{bmatrix}$

For example:

$5\begin{bmatrix}6 & 1 & 0\\-1 & 2 & 3\\0 & 1 & -4\end{bmatrix}=\begin{bmatrix}5.6 & 5.1 & 5.0\\5.-1 & 5.2 & 5.3\\5.0 & 5.1 & 5.-4\end{bmatrix}$

$=\begin{bmatrix}30 & 5 & 0\\-5 & 10 & 15\\0 & 5 & -20\end{bmatrix}$

Multiplying two matrices

Matrices can only be mulitplied together if the number of columns in the first matrix equals the number of rows in the second matrix.

Example:

Given two matrices $\mathit{A}=\begin{bmatrix}a_{1} & a_{2} & a_{3}\\a_{4} & a_{5} & a_{6} \\a_{7} & a_{8} & a_{9}\end{bmatrix}$ and $\mathit{B}=\begin{bmatrix}b_{1} & b_{2} & b_{3}\\b_{4} & b_{5} & b_{6} \\b_{7} & b_{8} & b_{9}\end{bmatrix}$

The product $\mathit{AB}=\begin{bmatrix}a_{1} & a_{2} & a_{3}\\a_{4} & a_{5} & a_{6} \\a_{7} & a_{8} & a_{9}\end{bmatrix}\begin{bmatrix}b_{1} & b_{2} & b_{3}\\b_{4} & b_{5} & b_{6} \\b_{7} & b_{8} & b_{9}\end{bmatrix}$