Notes On Minors and Cofactors - CBSE Class 12 Maths
Consider the determinant of square matrix A = [aij]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 R1, R2 and R3, and expanding it along columns C1, C2 and C3. Expanding the determinant along R1 ( 1st row): |A| = ${\text{(-1)}}^{\text{1+1}}\text{}{\text{a}}_{\text{11}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+2}}\text{}{\text{a}}_{\text{12}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+3}}\text{}{\text{a}}_{\text{13}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$ 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 aij of the determinant of matrix A can be obtained by finding the determinant of the matrix obtained by deleting the ith row and the jth column. The minor of an element aij is denoted by Mij. 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| = ${\text{(-1)}}^{\text{1+1}}\text{}{\text{a}}_{\text{11}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+2}}\text{}{\text{a}}_{\text{12}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+3}}\text{}{\text{a}}_{\text{13}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$ |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 aij be an element in the ith row and the jth column of a square matrix, A. Cofactor of aij  = (-1)i+j × Mij The cofactor of the element aij is denoted by Aij. |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| = ${\text{(-1)}}^{\text{1+2}}\text{}{\text{a}}_{\text{12}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{2+2}}\text{}{\text{a}}_{\text{22}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{3+2}}\text{}{\text{a}}_{\text{32}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$ ⇒ |A| =  +  – ${\text{}}^{\text{}}\text{}{\text{a}}_{\text{32}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$ 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.

#### Summary

Consider the determinant of square matrix A = [aij]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 R1, R2 and R3, and expanding it along columns C1, C2 and C3. Expanding the determinant along R1 ( 1st row): |A| = ${\text{(-1)}}^{\text{1+1}}\text{}{\text{a}}_{\text{11}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+2}}\text{}{\text{a}}_{\text{12}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+3}}\text{}{\text{a}}_{\text{13}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$ 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 aij of the determinant of matrix A can be obtained by finding the determinant of the matrix obtained by deleting the ith row and the jth column. The minor of an element aij is denoted by Mij. 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| = ${\text{(-1)}}^{\text{1+1}}\text{}{\text{a}}_{\text{11}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+2}}\text{}{\text{a}}_{\text{12}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{1+3}}\text{}{\text{a}}_{\text{13}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{21}}& {\text{a}}_{\text{22}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{32}}\end{array}\right|$ |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 aij be an element in the ith row and the jth column of a square matrix, A. Cofactor of aij  = (-1)i+j × Mij The cofactor of the element aij is denoted by Aij. |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| = ${\text{(-1)}}^{\text{1+2}}\text{}{\text{a}}_{\text{12}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{22}}& {\text{a}}_{\text{23}}\\ {\text{a}}_{\text{32}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{2+2}}\text{}{\text{a}}_{\text{22}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{31}}& {\text{a}}_{\text{33}}\end{array}\right|$ +  ${\text{(-1)}}^{\text{3+2}}\text{}{\text{a}}_{\text{32}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$ ⇒ |A| =  +  – ${\text{}}^{\text{}}\text{}{\text{a}}_{\text{32}}\text{}\left|\begin{array}{cc}{\text{a}}_{\text{11}}& {\text{a}}_{\text{13}}\\ {\text{a}}_{\text{21}}& {\text{a}}_{\text{23}}\end{array}\right|$ 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.

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