Let $A$ be a square matrix of order $3$, and their entries are denoted by $e_{11}$, $e_{12}$, $e_{13}$, $e_{21}$, $e_{22}$, $e_{23}$, $e_{31}$, $e_{32}$ and $e_{33}$ in algebraic form.
$A$ $\,=\,$ $\begin{bmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \\ \end{bmatrix}$
Now, let’s learn how to find the cofactors of the elements in a $3 \times 3$ square matrix by the concept of the minors of entries in a three by three square matrix.
The $e_{11}$, $e_{12}$ and $e_{13}$ are the entries in the first row of a $3 \times 3$ matrix $A$, and their corresponding cofactors are denoted by $C_{11}$, $C_{12}$ and $C_{13}$.
The $e_{11}$ is an entry in first row and first column of a $3×3$ matrix.
Now, calculate the product of the above two factors to find the cofactor $C_{11}$ mathematically.
$C_{11}$ $\,=\,$ $(-1)^{1+1} \times M_{11}$
$\implies$ $C_{11}$ $\,=\,$ $(-1)^{2} \times M_{11}$
$\implies$ $C_{11}$ $\,=\,$ $1 \times M_{11}$
$\implies$ $C_{11}$ $\,=\,$ $M_{11}$
$\implies$ $C_{11}$ $\,=\,$ $\begin{vmatrix} e_{22} & e_{23} \\ e_{32} & e_{33} \\ \end{vmatrix}$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{11}$ $\,=\,$ $e_{22} \times e_{33}$ $-$ $e_{32} \times e_{23}$
The $e_{12}$ is an element in first row and second column of a matrix of order $3$.
Now, evaluate the product of the above two factors to calculate the cofactor $C_{12}$ in mathematics.
$C_{12}$ $\,=\,$ $(-1)^{1+2} \times M_{12}$
$\implies$ $C_{12}$ $\,=\,$ $(-1)^{3} \times M_{12}$
$\implies$ $C_{12}$ $\,=\,$ $(-1) \times M_{12}$
$\implies$ $C_{12}$ $\,=\,$ $-M_{12}$
$\implies$ $C_{12}$ $\,=\,$ $-\begin{vmatrix} e_{21} & e_{23} \\ e_{31} & e_{33} \\ \end{vmatrix}$
$\implies$ $C_{12}$ $\,=\,$ $-(e_{21} \times e_{33}$ $-$ $e_{31} \times e_{23})$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{12}$ $\,=\,$ $e_{31} \times e_{23}$ $-$ $e_{21} \times e_{33}$
The $e_{13}$ is an entry in first row and third column of a $3×3$ matrix $A$.
Now, calculate the product of the above two factors to find the cofactor $C_{13}$ in mathematics.
$C_{13}$ $\,=\,$ $(-1)^{1+3} \times M_{13}$
$\implies$ $C_{13}$ $\,=\,$ $(-1)^{4} \times M_{13}$
$\implies$ $C_{13}$ $\,=\,$ $1 \times M_{13}$
$\implies$ $C_{13}$ $\,=\,$ $M_{13}$
$\implies$ $C_{13}$ $\,=\,$ $\begin{vmatrix} e_{21} & e_{22} \\ e_{31} & e_{32} \\ \end{vmatrix}$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{13}$ $\,=\,$ $e_{21} \times e_{32}$ $-$ $e_{31} \times e_{22}$
The $e_{21}$, $e_{22}$ and $e_{23}$ are the elements in the second row of a matrix $A$, and the cofactors of them are denoted by $C_{21}$, $C_{22}$ and $C_{23}$ respectively.
The $e_{21}$ is an element in second row and first column of a matrix of order three.
Now, find the product of the above two factors to evaluate the cofactor $C_{21}$ in mathematics.
$C_{21}$ $\,=\,$ $(-1)^{2+1} \times M_{21}$
$\implies$ $C_{21}$ $\,=\,$ $(-1)^{3} \times M_{21}$
$\implies$ $C_{21}$ $\,=\,$ $(-1) \times M_{21}$
$\implies$ $C_{21}$ $\,=\,$ $-M_{21}$
$\implies$ $C_{21}$ $\,=\,$ $-\begin{vmatrix} e_{12} & e_{13} \\ e_{32} & e_{33} \\ \end{vmatrix}$
$\implies$ $C_{21}$ $\,=\,$ $-(e_{12} \times e_{33}$ $-$ $e_{32} \times e_{13})$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{21}$ $\,=\,$ $e_{32} \times e_{13}$ $-$ $e_{12} \times e_{33}$
The $e_{22}$ is an entry in second row and second column of a $3 \times 3$ matrix $A$.
Now, find the product of the above two factors to find the cofactor $C_{22}$ mathematically.
$C_{22}$ $\,=\,$ $(-1)^{2+2} \times M_{22}$
$\implies$ $C_{22}$ $\,=\,$ $(-1)^{4} \times M_{22}$
$\implies$ $C_{22}$ $\,=\,$ $1 \times M_{22}$
$\implies$ $C_{22}$ $\,=\,$ $M_{22}$
$\implies$ $C_{22}$ $\,=\,$ $\begin{vmatrix} e_{11} & e_{13} \\ e_{31} & e_{33} \\ \end{vmatrix}$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{22}$ $\,=\,$ $e_{11} \times e_{33}$ $-$ $e_{31} \times e_{13}$
The $e_{23}$ is an element in second row and third column of a three by three matrix $A$.
Now, evaluate the product of the above two factors to calculate the cofactor $C_{23}$ in mathematics.
$C_{23}$ $\,=\,$ $(-1)^{2+3} \times M_{23}$
$\implies$ $C_{23}$ $\,=\,$ $(-1)^{5} \times M_{23}$
$\implies$ $C_{23}$ $\,=\,$ $(-1) \times M_{23}$
$\implies$ $C_{23}$ $\,=\,$ $-M_{23}$
$\implies$ $C_{23}$ $\,=\,$ $-\begin{vmatrix} e_{11} & e_{12} \\ e_{31} & e_{32} \\ \end{vmatrix}$
$\implies$ $C_{23}$ $\,=\,$ $-(e_{11} \times e_{32}$ $-$ $e_{31} \times e_{12})$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{23}$ $\,=\,$ $e_{31} \times e_{12}$ $-$ $e_{11} \times e_{32}$
The $e_{31}$, $e_{32}$ and $e_{33}$ are the elements in the third row of a square matrix $A$ of order $3$, and the cofactors are denoted by $C_{31}$, $C_{32}$ and $C_{33}$ respectively.
The $e_{31}$ is an entry in third row and first column of a square matrix of order $3$.
Now, evaluate the product of the above two factors to calculate the cofactor $C_{31}$ mathematically.
$C_{31}$ $\,=\,$ $(-1)^{3+1} \times M_{31}$
$\implies$ $C_{31}$ $\,=\,$ $(-1)^{4} \times M_{31}$
$\implies$ $C_{31}$ $\,=\,$ $1 \times M_{31}$
$\implies$ $C_{31}$ $\,=\,$ $M_{31}$
$\implies$ $C_{31}$ $\,=\,$ $\begin{vmatrix} e_{12} & e_{13} \\ e_{22} & e_{23} \\ \end{vmatrix}$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{31}$ $\,=\,$ $e_{12} \times e_{23}$ $-$ $e_{22} \times e_{13}$
The $e_{32}$ is an element in third row and second column of a $3 \times 3$ square matrix.
Now, find the product of the above two factors to find the cofactor $C_{32}$ mathematically.
$C_{32}$ $\,=\,$ $(-1)^{3+2} \times M_{32}$
$\implies$ $C_{32}$ $\,=\,$ $(-1)^{5} \times M_{32}$
$\implies$ $C_{32}$ $\,=\,$ $(-1) \times M_{32}$
$\implies$ $C_{32}$ $\,=\,$ $-M_{32}$
$\implies$ $C_{32}$ $\,=\,$ $-\begin{vmatrix} e_{11} & e_{13} \\ e_{21} & e_{23} \\ \end{vmatrix}$
$\implies$ $C_{32}$ $\,=\,$ $-(e_{11} \times e_{23}$ $-$ $e_{21} \times e_{13})$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{32}$ $\,=\,$ $e_{21} \times e_{13}$ $-$ $e_{11} \times e_{23}$
The $e_{33}$ is an element in third row and third column of a $3×3$ square matrix $A$.
Now, evaluate the product of the above two factors to find the cofactor $C_{33}$ in mathematics.
$C_{33}$ $\,=\,$ $(-1)^{3+3} \times M_{33}$
$\implies$ $C_{33}$ $\,=\,$ $(-1)^{6} \times M_{33}$
$\implies$ $C_{33}$ $\,=\,$ $1 \times M_{33}$
$\implies$ $C_{33}$ $\,=\,$ $M_{33}$
$\implies$ $C_{33}$ $\,=\,$ $\begin{vmatrix} e_{11} & e_{12} \\ e_{21} & e_{22} \\ \end{vmatrix}$
$\,\,\,\therefore\,\,\,\,\,\,$ $C_{33}$ $\,=\,$ $e_{11} \times e_{22}$ $-$ $e_{21} \times e_{12}$
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