Question

If $\Delta = \left| {\begin{array}{*{20}{c}}
  5&3&8 \\
  2&0&1 \\
  1&2&3
\end{array}} \right|$ , write the cofactor of the element ${a_{32}}$ .

Answer 1

Hint: First find the minor of the given element ${a_{32}}$ . The minor of the given element ${a_{32}}$can be found out by eliminating the elements of the second column and third row which gives the new element as $\left| {\begin{array}{*{20}{c}}
  5&8 \\
  2&1
\end{array}} \right|$. Now the cofactor of the given matrix can be found by the equation ${a_{32}} = {\left( { - 1} \right)^{3 + 2}}{M_{32}}$ where ${M_{32}}$ is the minor of the element ${a_{32}}$.

Complete step by step answer:
We have been given that $\Delta = \left| {\begin{array}{*{20}{c}}
  5&3&8 \\
  2&0&1 \\
  1&2&3
\end{array}} \right|$
Now first we will consider that ${M_{32}}$ represents the minor of the element ${a_{32}}$.
The minor of the given element ${a_{32}}$can be found out by eliminating the elements of the second column and third row.
By elimination of the second column and third row, we get the remaining elements as
${M_{32}}$$ = \left| {\begin{array}{*{20}{c}}
  5&8 \\
  2&1
\end{array}} \right|$
Now we can consider the term ${M_{32}}$ which can further be expressed as ${M_{32}}$= $5 - 16$ $ = - 11$ .
We know that the cofactor of any given matrix can be expressed as
${( - 1)^{m + n}}\left[ {Minor} \right]$
Where m represent the row number and n represent the column number respectively
Therefore we can express the cofactor of the given element by
Cofactor of ${a_{32}}$$ = {( - 1)^{3 + 2}}{M_{32}}$
Hence we get the cofactor of ${a_{32}}$ = ${\left( { - 1} \right)^{3 + 2}} = - 1$ $\left| {\begin{array}{*{20}{c}}
  5&8 \\
  2&1
\end{array}} \right|$
$ = - \left( { - 11} \right) = 11.$
Therefore the cofactor of the given matrix = 11.
Hence , we get the cofactor of the given matrix as 11.

Note:
The student must be careful in eliminating the column and rows to find out the necessary minor and cofactor of the given element. The student must also be careful in putting the desired values of m and n in the formula of the cofactor matrix which is given by ${\left( { - 1} \right)^{m + n}}\left[ {Minor} \right]$ where m represent the row number and n represent the column number respectively. If the m and the n values in the equation are not substituted properly, change of sign may occur which ultimately gives the wrong answer.