${\begin{vmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{vmatrix}}$ $\,=\,$ $e_{11} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,-\,$ $e_{12} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{23} \\ \end{vmatrix}}$
According to the definition of the determinant of a matrix, a formula for the determinant of a 3 by 3 matrix can be derived in algebraic form by following four fundamental steps. The following mathematical expression represents the determinant of a square matrix of the order $3$ in algebraic form.
${\begin{vmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{vmatrix}}$
Now, let’s learn how to derive the determinant formula for the matrix of the order $3 \times 3$ by following the four steps.
$\left|\begin{array}{c | c c} \color{red} e_{11} & e_{12} & e_{13} \\ \hline e_{21} & \color{blue} e_{22} & \color{blue} e_{23} \\ e_{31} & \color{blue} e_{32} & \color{blue} e_{33} \\ \end{array}\right|$
$\implies$ $\left|\begin{array}{c | c c} \color{red} e_{11} & e_{12} & e_{13} \\ \hline e_{21} & \color{blue} e_{22} & \color{blue} e_{23} \\ e_{31} & \color{blue} e_{32} & \color{blue} e_{33} \\ \end{array}\right|$ $\,=\,$ $e_{11} \times (-1)^{1+1} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$
$\left|\begin{array}{c|c|c} e_{11} & \color{red} e_{12} & e_{13} \\ \hline \color{blue} e_{21} & e_{22} & \color{blue} e_{23} \\ \color{blue} e_{31} & e_{32} & \color{blue} e_{33} \\ \end{array}\right|$
$\implies$ $\left|\begin{array}{c|c|c} e_{11} & \color{red} e_{12} & e_{13} \\ \hline \color{blue} e_{21} & e_{22} & \color{blue} e_{23} \\ \color{blue} e_{31} & e_{32} & \color{blue} e_{33} \\ \end{array}\right|$ $\,=\,$ $e_{12} \times (-1)^{1+2} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$
$\left|\begin{array}{cc|c} e_{11} & e_{12} & \color{red} e_{13} \\ \hline \color{blue} e_{21} & \color{blue} e_{22} & e_{23} \\ \color{blue} e_{31} & \color{blue} e_{32} & e_{33} \\ \end{array}\right|$
$\implies$ $\left|\begin{array}{cc|c} e_{11} & e_{12} & \color{red} e_{13} \\ \hline \color{blue} e_{21} & \color{blue} e_{22} & e_{23} \\ \color{blue} e_{31} & \color{blue} e_{32} & e_{33} \\ \end{array}\right|$ $\,=\,$ $e_{13} \times (-1)^{1+3} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{23} \\ \end{vmatrix}}$
Finally, add the products for calculating the determinant of a square matrix of the order $3$.
${\begin{vmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{vmatrix}}$ $\,=\,$ $e_{11} \times (-1)^{1+1} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{12} \times (-1)^{1+2} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times (-1)^{1+3} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}}$
Now, let’s evaluate the product in each term in the right hand side expression.
$=\,\,\,$ $e_{11} \times (-1)^{2} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{12} \times (-1)^{3} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times (-1)^{4} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}}$
$=\,\,\,$ $e_{11} \times 1 \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{12} \times (-1) \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times 1 \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}}$
$\therefore\,\,\,$ ${\begin{vmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{vmatrix}}$ $\,=\,$ $e_{11} \times {\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}}$ $\,-\,$ $e_{12} \times {\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}}$ $\,+\,$ $e_{13} \times {\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}}$
It can used as a formula for calculating the determinant of a third order square matrix. It can also be simplified further by the determinant of a second order matrix.
$=\,\,\,$ $e_{11} \times \Big(e_{22} \times e_{33}-e_{23} \times e_{32}\Big)$ $\,-\,$ $e_{12} \times \Big(e_{21} \times e_{33}-e_{23} \times e_{31}\Big)$ $\,+\,$ $e_{13} \times \Big(e_{21} \times e_{32}-e_{22} \times e_{31}\Big)$
$=\,\,\,$ $e_{11} \times \Big(e_{22}e_{33}-e_{23}e_{32}\Big)$ $\,-\,$ $e_{12} \times \Big(e_{21}e_{33}-e_{23}e_{31}\Big)$ $\,+\,$ $e_{13} \times \Big(e_{21}e_{32}-e_{22}e_{31}\Big)$
Each multiplying factor can be distributed to the difference of the terms in the expression by the distributive property of multiplication over subtraction.
$=\,\,\,$ $e_{11} \times e_{22}e_{33}$ $\,-\,$ $e_{11} \times e_{23}e_{32}$ $\,-\,$ $e_{12} \times e_{21}e_{33}$ $\,+\,$ $e_{12} \times e_{23}e_{31}$ $\,+\,$ $e_{13} \times e_{21}e_{32}$ $\,-\,$ $e_{13} \times e_{22}e_{31}$
$=\,\,\,$ $e_{11}e_{22}e_{33}$ $\,-\,$ $e_{11}e_{23}e_{32}$ $\,-\,$ $e_{12}e_{21}e_{33}$ $\,+\,$ $e_{12}e_{23}e_{31}$ $\,+\,$ $e_{13}e_{21}e_{32}$ $\,-\,$ $e_{13}e_{22}e_{31}$
$\therefore\,\,\,$ ${\begin{vmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{vmatrix}}$ $\,=\,$ $e_{11}e_{22}e_{33}$ $\,+\,$ $e_{12}e_{23}e_{31}$ $\,+\,$ $e_{13}e_{21}e_{32}$ $\,-\,$ $e_{11}e_{23}e_{32}$ $\,-\,$ $e_{12}e_{21}e_{33}$ $\,-\,$ $e_{13}e_{22}e_{31}$
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