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Solving a 3 x 3 System of Simultaneous Equation Using Matrix Techniques

Suppose we have a system of equations
\[a_{11} x + a_{12}y + a_{13} z= \alpha\]

\[a_{21} x + a_{22}y + a_{23} z= \beta\]

\[a_{31} x + a_{32}y + a_{33} z= \gamma\]

We can write this in matrix form as
\[ \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right) \begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix}\]

Let  
\[\mathbf{a} = \begin{pmatrix}a_{11}\\a_{21}\\a_{31}\end{pmatrix}, \: \mathbf{b} = \begin{pmatrix}a_{12}\\a_{22}\\a_{32}\end{pmatrix} , \: \mathbf{c} = \begin{pmatrix}a_{13}\\a_{23}\\a_{33}\end{pmatrix}\]

Then  
\[A^{-1} = \frac{1}{det (A)} ( \mathbf{b} \times \mathbf{c} \: \mathbf{c} \times \mathbf{a} \; \mathbf{a} \times \mathbf{b} ) \]
.
Hence  
\[\begin{pmatrix}x\\y\\z\end{pmatrix} =\frac{1}{det (A)} ( \mathbf{b} \times \mathbf{c} \: \mathbf{c} \times \mathbf{a} \; \mathbf{a} \times \mathbf{b} ) \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} = \begin{pmatrix} \frac{( \mathbf{b} \times \mathbf{c} ) \cdot (\alpha \mathbf{i} + \beta \mathbf{j} + \gamma \mathbf{j})}{det(A)} \\ \frac{( \mathbf{c} \times \mathbf{a} ) \cdot (\alpha \mathbf{i} + \beta \mathbf{j} + \gamma \mathbf{j})}{det(A)} \\ \frac{( \mathbf{a} \times \mathbf{b} ) \cdot (\alpha \mathbf{i} + \beta \mathbf{j} + \gamma \mathbf{j})}{det(A)} \end{pmatrix} \]
.
In matrix form
\[x= \frac{\left| \begin{array}{ccc} \alpha & a_{12} & a_{13} \\ \beta & a_{22} & a_{23} \\ \gamma & a_{32} & a_{33} \end{array} \right|}{\left| \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right|} , \:y= \frac{\left| \begin{array}{ccc} a_{11} & \alpha & a_{13} \\ a_{21} & \beta & a_{23} \\ a_{31} & \gamma & a_{33} \end{array} \right|}{\left| \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right|} , \:z= \frac{\left| \begin{array}{ccc} a_{11} & a_{12} & \alpha \\ a_{21} & a_{22} & \beta \\ a_{31} & a_{32} & \gamma \end{array} \right|}{\left| \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right|}\]
.