Question

How to find all the minors and cofactors of the matrix \[A = \left( {\begin{array}{*{20}{c}}1&{ - 2}&3\\6&7&{ - 1}\\{ - 3}&1&4\end{array}} \right)\]?

Answer 1

Hint: For the matrix \[A = {\left[ {{a_{ij}}} \right]_{n \times n}}\] of order \[n\], with elements represented by term \[{a_{ij}}\] such that \[1 \le i,j \le n\]; the minor for any element \[{a_{ij}}\] is the determinant of the matrix formed by deleting the \[{i^{th}}\] row and \[{j^{th}}\] column.

The cofactor of any element \[{a_{ij}}\] for the matrix \[A = {\left[ {{a_{ij}}} \right]_{n \times n}}\] is equal to \[{\left( { - 1} \right)^{i + j}}{M_{ij}}\]. Here \[{M_{ij}}\]is the minor of the matrix \[A\]corresponding to element \[{a_{ij}}\].

Complete step by step solution:
The given square matrix \[A\] can be written as shown below.
\[A = \left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\{{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\{{a_{31}}}&{{a_{32}}}&{{a_{33}}}\end{array}} \right)\]
\[ = \left( {\begin{array}{*{20}{c}}1&{ - 2}&3\\6&7&{ - 1}\\{ - 3}&1&4\end{array}} \right)\]

Let’s start to find minor \[{M_{11}}\] corresponding to element \[{a_{11}} = 1\] as follows:

Delete the first row and first column of the matrix \[A\] where the element \[{a_{11}} = 1\] is situated and obtains the determinant of the matrix formed in this process.
\[{M_{11}} = \left| {\begin{array}{*{20}{c}}7&{ - 1}\\1&4\end{array}} \right|\]

Evaluate the determinant as shown below.
\[{M_{11}} = \left( 7 \right)\left( 4 \right) - \left( 1 \right)\left( { - 1} \right)\]
\[ = 28 + 1\]
\[ = 29\]
Therefore, the minor of the element \[{a_{11}} = 1\] for the matrix \[A\] is \[{M_{11}} = 29\].

now, use the relation between cofactor and minor which is \[{C_{ij}} = {\left( { - 1} \right)^{i + j}}{M_{ij}}\] to obtain the cofactor \[{C_{11}}\] corresponding to element \[{a_{11}} = 1\] for the matrix \[A\] is shown below.
\[{C_{11}} = {\left( { - 1} \right)^{1 + 1}}{M_{11}}\]
\[ = {\left( { - 1} \right)^2}\left( {29} \right)\]
\[ = 29\]
Therefore, corresponding to \[{a_{11}} = 1\] for the matrix \[A\] the minor is \[{M_{11}} = 29\] and the cofactor is \[{C_{11}} = 29\].

Similarly, obtain other minors and cofactors corresponding to each element of matrix \[A\].

Delete the first row and second column of the matrix \[A\] and obtain the determinant of the matrix formed in this process.
\[{M_{12}} = \left| {\begin{array}{*{20}{c}}6&{ - 1}\\{ - 3}&4\end{array}} \right|\]

Evaluate the determinant as shown below.
\[{M_{12}} = \left( 6 \right)\left( 4 \right) - \left( { - 3} \right)\left( { - 1} \right)\]
\[ = 24 - 3\]
\[ = 21\]
Therefore, the minor of the element \[{a_{12}} = - 2\] for the matrix \[A\] is \[{M_{12}} = 21\].

The cofactor \[{C_{12}}\] corresponding to element \[{a_{12}} = - 2\] for the matrix \[A\] is shown below.
\[{C_{12}} = {\left( { - 1} \right)^{1 + 2}}{M_{12}}\]
\[ = {\left( { - 1} \right)^3}\left( {21} \right)\]
\[ = - 21\]

Therefore, corresponding to \[{a_{12}} = - 2\] for the matrix \[A\] the minor is \[{M_{12}} = 21\] and the cofactor is \[{C_{12}} = - 21\].

The complete table for each minor and cofactor corresponding to each element from matrix \[A\] is shown below.

\[{a_{ij}}\]	\[{M_{ij}}\]	\[{C_{ij}}\]
\[{a_{11}} = 1\]	\[29\]	\[29\]
\[{a_{12}} = - 2\]	\[21\]	\[ - 21\]
\[{a_{13}} = 3\]	\[27\]	\[27\]
\[{a_{21}} = 6\]	\[ - 11\]	\[11\]
\[{a_{22}} = 7\]	\[13\]	\[13\]
\[{a_{23}} = - 1\]	\[ - 5\]	\[5\]
\[{a_{31}} = - 3\]	\[ - 19\]	\[ - 19\]
\[{a_{32}} = 1\]	\[ - 19\]	\[19\]
\[{a_{32}} = 4\]	\[19\]	\[19\]

Therefore, the minor matrix and cofactor matrix for matrix \[A\] is written as shown below.
\[ \Rightarrow M = \left( {\begin{array}{*{20}{c}}{29}&{21}&{27}\\{ - 11}&{13}&{ - 5}\\{ - 19}&{ - 19}&{19}\end{array}} \right)\]
\[ \Rightarrow C = \left( {\begin{array}{*{20}{c}}{29}&{ - 21}&{27}\\{11}&{13}&5\\{ - 19}&{19}&{19}\end{array}} \right)\]


Note: The cofactor and minor for an element at even position like \[{a_{11}}\], \[{a_{13}}\], \[{a_{22}}\] etc. are the same and at odd positions negative to each other. In other words, \[{C_{ij}} = {M_{ij}}\] if \[\left( {i + j} \right)\] is even and \[{C_{ij}} = - {M_{ij}}\] if \[\left( {i + j} \right)\] is odd where \[{a_{ij}}\] is the element.