Consider the determinant of square matrix A = [a_ij]_3x3.

i.e. |A| = $\left|\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$

The determinant of matrix A can be calculated in 6 ways. They are expanding the determinant along rows R_1, R₂ and R₃, and expanding it along columns C₁, C₂ and C₃.

Expanding the determinant along R₁ ( 1^st row):

|A| = ${\text{(-1)}}^{\text{1+1}}{\text{a}}_{\text{11}}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+2}}{\text{a}}_{\text{12}}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+3}}{\text{a}}_{\text{13}}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$

Multiply the first element, a₁₁, by (-1)¹⁺¹ and the second order determinant obtained by deleting the elements of the first row and the first column of matrix A.

The minor of an element a_ij of the determinant of matrix A can be obtained by finding the determinant of the matrix obtained by deleting the i^th row and the j^th column.

The minor of an element a_ij is denoted by M_ij.

The minor of a₁₂ can be obtained by finding the determinant of the matrix obtained by deleting the first row and the second column.

Minor of element a₁₂ = $\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$

Minor of element a₁₃ = $\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$

|A| = ${\text{(-1)}}^{\text{1+1}}{\text{a}}_{\text{11}}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+2}}{\text{a}}_{\text{12}}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+3}}{\text{a}}_{\text{13}}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$

|A| = (-1)¹⁺¹a₁₁M₁₁ + (-1)¹⁺²a₁₂M₁₂ + (-1)¹⁺³a₁₃M₁₃

The minor of an element of a determinant of order n(n ≥ 2) is a determinant of order (n - 1).

Cofactor:

Let a_ij be an element in the i^th row and the j^th column of a square matrix, A.

Cofactor of a_ij  = (-1)^i+j × M_ij

The cofactor of the element a_ij is denoted by A_ij.

|A| = a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃

	|A|
Expansion along 2nd row	= a21A21 + a22A22 + a23A23
Expansion along 3rd row	= a31A31 + a32A32 + a33A33
Expansion along 1st column	= a11A11 + a21A21 + a31A31
Expansion along 2nd column	= a12A12 + a22A22 + a32A32
Expansion along 3rd column	= a13A13 + a23A23 + a33A33

The determinant of a matrix is equal to the sum of the products of the elements of any row or column with their corresponding cofactors.

A = $\left[\begin{array}{ccc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{12}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right]$

Minor of a₁₂, M₁₂ = $\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{23}}\end{array}\right|$

Minor of a₂₂, M₂₂ = $\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$

Minor of a₃₂, M₃₂ = $\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$

Cofactor of a₁₂, A₁₂ = (–1)¹⁺²  $\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{23}}\end{array}\right|$

Cofactor of a₂₂, A₂₂ = (–1)²⁺²  $\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{23}}\end{array}\right|$

Cofactor of a₃₂, A₃₂ = (–1)³⁺²  $\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$

The determinant of a matrix is equal to the sum of the products of the elements of a row or a column with their corresponding cofactors.

∴ |A| = a₁₂A₁₂ + a₂₂A₂₂ + a₃₂A₃₂

|A| = ${\text{(-1)}}^{\text{1+2}}{\text{a}}_{\text{12}}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{2+2}}{\text{a}}_{\text{22}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{3+2}}{\text{a}}_{\text{32}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$

⇒ |A| = ${\text{–}}^{}{\text{a}}_{\text{12}}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${}^{}{\text{a}}_{\text{22}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$– ${}^{}{\text{a}}_{\text{32}}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$

Note

If the elements of a row (or column) are multiplied with the cofactors of any other row (or column), then their sum is zero.