Question

How do I find the determinant of a $3\times 3$ matrix?

Answer 1

Hint: The determinant is a scalar value that can be defined as the function of the entries of a square matrix. A determinant of a given matrix can be calculated by taking its minor and then simplifying each minor term.

Complete step by step solution:
Matrices are the ordered rectangular array of numbers that are used to express linear equations, which include rows and columns. There are many different types of matrices, which depend on various factors such as the numbers of rows and columns or the entries in the matrix.
A $3\times 3$ matrix can be given as $A=\left[ \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\
   {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\
   {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right]$
The determinant of this matrix can be calculated by taking its minor along the first row and then further simplifying the $2\times 2$ matrices by taking their minors and simplifying them.
The determinant of the above given matrix is represented by $\left| A \right|$
Therefore, the determinant of the above matrix can be written as:
 $\left| A \right|=\left[ \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\
   {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\
   {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right]={{a}_{11}}\left| \begin{matrix}
   {{a}_{22}} & {{a}_{23}} \\
   {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right|-{{a}_{12}}\left| \begin{matrix}
   {{a}_{21}} & {{a}_{23}} \\
   {{a}_{31}} & {{a}_{33}} \\
\end{matrix} \right|+{{a}_{13}}\left| \begin{matrix}
   {{a}_{21}} & {{a}_{22}} \\
   {{a}_{31}} & {{a}_{32}} \\
\end{matrix} \right|$ …$(i)$
The above expression can be further simplified by calculating the determinants of the $2\times 2$ matrices and then simply adding or subtracting the values obtained by simplifying them.
A $2\times 2$ matrix can be solved in the following manner:
$\left| \begin{matrix}
   p & q \\
   r & s \\
\end{matrix} \right|=p\times s-q\times r$ Applying the same concept in the equation $(i)$ we get:
$\Rightarrow \left| A \right|=\left[ \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\
   {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\
   {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right]={{a}_{11}}({{a}_{22}}\times {{a}_{33}}-{{a}_{23}}\times {{a}_{32}})-{{a}_{12}}({{a}_{21}}\times {{a}_{33}}-{{a}_{23}}\times {{a}_{31}})+{{a}_{13}}({{a}_{21}}\times {{a}_{32}}-{{a}_{22}}\times {{a}_{31}})$By simplifying the above expression, we can find the scalar value of the determinant.
Hence, the determinant of a $3\times 3$matrix can be calculated in the following way:
$\left| A \right|=\left[ \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\
   {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\
   {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right]={{a}_{11}}({{a}_{22}}\times {{a}_{33}}-{{a}_{23}}\times {{a}_{32}})-{{a}_{12}}({{a}_{21}}\times {{a}_{33}}-{{a}_{23}}\times {{a}_{31}})+{{a}_{13}}({{a}_{21}}\times {{a}_{32}}-{{a}_{22}}\times {{a}_{31}})$

Note: The above mentioned method to calculate the determinant of a matrix is known as the Laplace Expansion. There are many other ways to solve and calculate the value of the determinants.
On the other hand, determinant can be defined as the scalar value of a matrix and can be denoted by $\det (A)$, $\det A$ or $\left| A \right|$.