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Matrices

Solving system of equations using the inverse matrix method
The inverse matrix method uses the inverse of a matrix to help solve a system of equations.
Consider the simultaneous equations:

Provided you understand how matrices are multiplied together you will realise that these can
be written in matrix form as:

writing:

A=, X= and B=

we have A·X=B.

Given AX = B we can multiply both sides by the inverse of A, provided this exists, to give:
A-1·A·X=A-1·B.
But A-1A = I, the identity matrix, so
X=A-1·B

$\fs2\left.\begin{matrix}-x+y=-5\\2x-3y=3\\-z=2\\nd{matrix}\right\}\Rightarrow\begin{pmatrix}-1&1&0\\2&-3&0\\0&0&-1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}-5\\3\\2\end{pmatrix}$
$\fs2A^{-1}=\begin{pmatrix}-3&-1&0\\-2&-1&0\\0&0&-1\end{pmatrix}\;Then\;X=\begin{pmatrix}-3&-1&0\\-2&-1&0\\0&0&-1\end{pmatrix}\begin{pmatrix}-5\\3\\2\end{pmatrix}\Rightarrow\;X=\begin{pmatrix}12\\7\\-2\end{pmatrix}$

Solve using the inverse matrix method

 $\left.\begin{matrix}-3x-2y+5z=-4\\x+3y-3z=5\\x+5y-4z=5\\nd{matrix}\right\}$ X =