Question

If we have a matrix as $\Delta =\left( \begin{matrix} 1 & 2 & 3 \\\ 2 & 0 & 1 \\\ 5 & 3 & 8 \\\ \end{matrix} \right)$, then write the minor of elements ${{a}_{22}}$

Answer 1

Firstly we will represent our given matrix into a general 3x3 matrix from where we will learn that where is ${{a}_{22}}$ and other similar elements then we will eliminate that row and that column in which ${{a}_{22}}$ is present then form a matrix out of remaining elements present in the matrix now after getting a 2x2 new matrix of remaining elements we just have to find the determinant of that 2x2 matrix and that determinant is the answer.

Complete step-by-step solution:
Given a 3x3 matrix $\Delta =\left( \begin{matrix} 1 & 2 & 3 \\\ 2 & 0 & 1 \\\ 5 & 3 & 8 \\\ \end{matrix} \right)$ and we are asked to find the minor of elements ${{a}_{22}}$
Firstly we can write any 3x3 matrix $\Delta =\left( \begin{matrix} 1 & 2 & 3 \\\ 2 & 0 & 1 \\\ 5 & 3 & 8 \\\ \end{matrix} \right)$ as $\left( \begin{matrix} 1 & 2 & 3 \\\ 2 & 0 & 1 \\\ 5 & 3 & 8 \\\ \end{matrix} \right)=\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\\ \end{matrix} \right)$ now we can clearly see all elements of matrix and there representation now we get that ${{a}_{22}}$ is the middle element which is 0
Second step now eliminate the row and elements column of ${{a}_{22}}$or 0 containing. So, after eliminating row and 0 containing column our matrix will look like

1 & {} & 3 \\\ {} & {} & {} \\\ 5 & {} & 8 \\\ \end{matrix} \right)$$, which is a 2x2 matrix, now we just have to find the determinant of this 2x2 matrix which is given by $$\det \left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)={{a}_{11}}\times {{a}_{22}}-{{a}_{12}}\times {{a}_{21}}$$ So, the determinant of our equation $$\left( \begin{matrix} 1 & {} & 3 \\\ {} & {} & {} \\\ 5 & {} & 8 \\\ \end{matrix} \right)$$ will be given as $$8\times 1-5\times 3=8-15$$ Which comes out be $$-7$$ **Hence minor of elements $${{a}_{22}}$$ is $$-7$$** **Note:** If in this case we have been given 2x2 matrix instead of 3x3 and now we are asked to find minor of the matrix, we will learn by taking an example $$\Delta =\left( \begin{matrix} 1 & 3 \\\ 4 & 2 \\\ \end{matrix} \right)$$ here if we have to find minor of $${{a}_{22}}$$ $$\Delta =\left( \begin{matrix} 1 & 3 \\\ 4 & 2 \\\ \end{matrix} \right)=\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)$$ now we got that $${{a}_{22}}=2$$ so after eliminating that row and that column in which $${{a}_{22}}$$ is present ,element remaining will be 1 hence 1 will be minor of $${{a}_{22}}$$ in matrix $$\Delta =\left( \begin{matrix} 1 & 3 \\\ 4 & 2 \\\ \end{matrix} \right)$$