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Question: If \(\Delta = \left| {\begin{array}{*{20}{c}} 5&3&8 \\\ 2&0&1 \\\ 1&2&3 \end{array}...

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 a32{a_{32}} .

Explanation

Solution

First find the minor of the given element a32{a_{32}} . The minor of the given element a32{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 a32=(−1)3+2M32{a_{32}} = {\left( { - 1} \right)^{3 + 2}}{M_{32}} where M32{M_{32}} is the minor of the element a32{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 M32{M_{32}} represents the minor of the element a32{a_{32}}.
The minor of the given element a32{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
M32{M_{32}} = \left| {\begin{array}{*{20}{c}} 5&8 \\\ 2&1 \end{array}} \right|
Now we can consider the term M32{M_{32}} which can further be expressed as M32{M_{32}}= 5−165 - 16 =−11 = - 11 .
We know that the cofactor of any given matrix can be expressed as
(−1)m+n[Minor]{( - 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 a32{a_{32}} =(−1)3+2M32 = {( - 1)^{3 + 2}}{M_{32}}
Hence we get the cofactor of a32{a_{32}} = (−1)3+2=−1{\left( { - 1} \right)^{3 + 2}} = - 1 \left| {\begin{array}{*{20}{c}} 5&8 \\\ 2&1 \end{array}} \right|
=−(−11)=11.= - \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 (−1)m+n[Minor]{\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.