Question

If the determinant of the matrix is given as $\Delta =\left| \begin{matrix} 5 & 3 & 8 \\\ 2 & 0 & 1 \\\ 1 & 2 & 3 \\\ \end{matrix} \right|$, then find the minor of the element ${{a}_{23}}$.

Answer 1

We start solving the problem by recalling the notation of elements in a matrix and calculation of minors of an element in the given matrix. We write the minor of the given element in the determinant form and calculate using the formula of determinant to find the required value.

Complete step by step answer:
According to the problem, we have given that the determinant of the matrix is given as $\Delta =\left| \begin{matrix} 5 & 3 & 8 \\\ 2 & 0 & 1 \\\ 1 & 2 & 3 \\\ \end{matrix} \right|$. We need to find the value of the minor of the element ${{a}_{23}}$ of the matrix.
We know that ${{a}_{23}}$ is the element that is present in the second $\left( {{2}^{nd}} \right)$ row and third $\left( {{3}^{rd}} \right)$ column of the matrix.
We know that minor of the element in a matrix is defined as the determinant of the matrix that is formed after deleting all the elements present in the row and column that the given element lies i.e., the minor of the element ${{a}_{23}}$ of the matrix $\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\\ \end{matrix} \right]$ is defined as $\left| \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{31}} & {{a}_{32}} \\\ \end{matrix} \right|$.
Let us assume the minor of the element ${{a}_{23}}$ of the matrix $\left| \begin{matrix} 5 & 3 & 8 \\\ 2 & 0 & 1 \\\ 1 & 2 & 3 \\\ \end{matrix} \right|$ as ‘m’.
We have got the value of $m=\left| \begin{matrix} 5 & 3 \\\ 1 & 2 \\\ \end{matrix} \right|$.
We know that the determinant of the matrix $\left| \begin{matrix} a & b \\\ c & d \\\ \end{matrix} \right|$ is defined as $\left( a\times d \right)-\left( b\times c \right)$.
We have got the value of $m=\left( 5\times 2 \right)-\left( 3\times 1 \right)$.
We have got the value of m = 10 – 3.
We have got the value of m = 7.
We have found the value of the minor of the element ${{a}_{23}}$ of the matrix $\left| \begin{matrix} 5 & 3 & 8 \\\ 2 & 0 & 1 \\\ 1 & 2 & 3 \\\ \end{matrix} \right|$ as 7.
∴ The value of the minor of the element ${{a}_{23}}$ of the matrix $\left| \begin{matrix} 5 & 3 & 8 \\\ 2 & 0 & 1 \\\ 1 & 2 & 3 \\\ \end{matrix} \right|$ is 7.

Note:
We should not confuse the notation of representation of an element in a matrix. We also should not multiply ${{\left( -1 \right)}^{2+3}}$ to ‘m’ as we are calculating the value of minor. If we are calculating the cofactor of that element, we multiply minor with ${{\left( -1 \right)}^{2+3}}$. Similarly, we can expect problems to solve for cofactor and determinant.