Question

If ${{A}_{ij}}$ is the cofactor of the element ${{a}_{ij}}$of the determinant $\left[ \begin{matrix}
   2 & -3 & 5 \\
   6 & 0 & 4 \\
   1 & 5 & -7 \\
\end{matrix} \right]$ then write the value of ${{a}_{32}}.{{A}_{32}}$

Answer 1

Hint: A cofactor is a number you get when you remove the column and row of a designated element in a matrix.
For a given matrix of order $3\times 3$; say \[A=\left[ \begin{matrix}
   {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\
   {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\
   {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\
\end{matrix} \right]\],
Cofactor of any element ${{a}_{ij}}$ is given as ${{A}_{ij}}={{\left( -1 \right)}^{i+j}}{{M}_{ij}}$
Where, ${{M}_{ij}}$ is the matrix formed by removing the ${{i}^{th}}$ row and ${{j}^{th}}$ column of the matrix. Use the above formulas to find the cofactor of ${{a}_{32}}$i.e. ${{A}_{32}}$ and get the value of ${{a}_{32}}.{{A}_{32}}$

Complete step-by-step solution:
Since we have a matrix of order $3\times 3$; $A=\left[ \begin{matrix}
   2 & -3 & 5 \\
   6 & 0 & 4 \\
   1 & 5 & -7 \\
\end{matrix} \right]$
So, element ${{a}_{32}}$ is 5.
Now, we need to find ${{M}_{32}}$first.
So, ${{M}_{32}}=\left[ \begin{matrix}
   2 & 5 \\
   6 & 4 \\
\end{matrix} \right]......(1)$
Now, solve the equation (1), we get:
$\begin{align}
  & \Rightarrow {{M}_{32}}=\left[ 8-30 \right] \\
 & \Rightarrow {{M}_{32}}=-22 \\
\end{align}$
Now, we need to calculate value of ${{A}_{32}}$
By using the formula ${{A}_{ij}}={{\left( -1 \right)}^{i+j}}{{M}_{ij}}$, where i=3 and j = 2, we get:
$\begin{align}
  & \Rightarrow {{A}_{32}}={{\left( -1 \right)}^{3+2}}\left( -22 \right) \\
 & \Rightarrow {{A}_{32}}=-1\times -22 \\
 & \Rightarrow {{A}_{32}}=22 \\
\end{align}$
Now substitute the values of ${{a}_{32}}$and ${{A}_{32}}$ and get the value of ${{a}_{32}}.{{A}_{32}}$.
We get:
$\begin{align}
  & {{a}_{32}}.{{A}_{32}}=5\times 22 \\
 & =110
\end{align}$
Hence, the value of ${{a}_{32}}.{{A}_{32}}$ is 110.

Note: Always remember that while finding the co-factor of an element, find the matrix formed by removing the elements of row and column of that element first. Elements of the matrix can be misplaced in hurry. So, be careful while finding ${{M}_{ij}}$. And to find the value of ${{M}_{ij}}$, expand the matrix formed to get a finite value.
Also, some might confuse with the definition of co-factor. So, instead of using the given formula to find co-factor${{A}_{ij}}={{\left( -1 \right)}^{i+j}}{{M}_{ij}}$, they might write ${{M}_{ij}}$ as co-factor of the element directly. It is an incomplete answer. Always follow the given formula to find the co-factor.