Minors and Co-factors in Matrix

Before learning how to find the determinant of a 3×3 matrix, we must learn about what is minors and co-factors in matrix. One reading the following article must have the knowledge of the following topics

• Sub-matrix
• Determinant of 2×2 Matrix

Background

Let us consider a matrix A of order 3×3 as

$\text{A} = \left ( \displaylines{a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}} \right )_{3\text{x}3}$

In short,

$\text{A} = \left ( a_{ij} \right)$

where i = 1,2,3 and j = 1,2,3.

Information obtained from matrix A

• A is a square matrix.
• Matrix A is of order 3×3. This means it has three rows and three columns.
• Each element of the matrix is denoted by a small letter (same alphabet used to denote the matrix) along with the identity of the row and column that the element belongs to. The number i denotes its row and the number j denotes its columns.

Above-mentioned terms had already been discussed in the first chapter – Introduction to Matrix.

Minors in Matrix

Let $\text{A} = (a_{ij})$ be a matrix of order mxm where i = 1,2,3,…,m and j = 1,2,3,…,m then minor of an element of A, say $(a_{ij})$, is defined as the 2×2 determinant of a square sub-matrix formed by deleting/removing the i^th row and j^th column of the matrix A. Here, minor of element $a_{ij}$ is denoted by $\text{M}_{ij}$.

Example: How to find the minor of element $a_{11}$ of a matrix A?

Considering the above-mentioned matrix A

$\text{A} = \left ( \displaylines{a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}} \right )_{3\text{x}3}$

STEP 1: Locate the position of the element in the matrix by the values of i and j.

For the element $a_{11}$, it is positioned in the 1^st row and the 1^st column of the matrix A.

STEP 2: Form a sub-matrix by eliminating the i^th row and j^th column from the parent matrix A.

For the element $a_{11}$, we need to form a sub-matrix by eliminating the 1^st row and the 1^st column of the matrix A. Let us denote this submatrix by letter $\text{M}_{11}$ where the base of the matrix signify the values of i and j. Hence,

$\text{M}_{11} = \left ( \displaylines{ a_{22} & a_{23} \\ a_{32} & a_{33} } \right )$

STEP 3: Find the determinant of the sub-matrix obtained in step 2.

$\text{M}_{11} = \left ( \displaylines{ a_{22} & a_{23} \\ a_{32} & a_{33} } \right )$

$\text{|M}_{11}| = \left | \displaylines{ a_{22} & a_{23} \\ a_{32} & a_{33} } \right |$

$\text{|M}_{11}| = (a_{22} \times a_{33} \ – \ a_{23} \times a_{32} )$

Hence, the minor of element $a_{11}$ of matrix A is given by

$\text{M}_{11} \ = \ (a_{22} \times a_{33} \ – \ a_{23} \times a_{32} )$

Similarly

$\text{M}_{12} \ = \ \left | \displaylines{ a_{21} & a_{23} \\ a_{31} & a_{33} } \right | $

$\text{M}_{13} \ = \ \left | \displaylines{ a_{21} & a_{22} \\ a_{31} & a_{32} } \right | $

$\text{M}_{21} \ = \ \left | \displaylines{ a_{12} & a_{13} \\ a_{32} & a_{33} } \right | $

$\text{M}_{22} \ = \ \left | \displaylines{ a_{11} & a_{13} \\ a_{31} & a_{33} } \right | $

$\text{M}_{23} \ = \ \left | \displaylines{ a_{11} & a_{12} \\ a_{31} & a_{32} } \right | $

$\text{M}_{31} \ = \ \left | \displaylines{ a_{12} & a_{13} \\ a_{22} & a_{23} } \right | $

$\text{M}_{32} \ = \ \left | \displaylines{ a_{11} & a_{13} \\ a_{21} & a_{23} } \right | $

$\text{M}_{33} \ = \ \left | \displaylines{ a_{11} & a_{12} \\ a_{21} & a_{22} } \right | $

Co-factors in Matrix

Let $\text{A} = (a_{ij})$ be a matrix of order mxm where i = 1,2,3,…,m and j = 1,2,3,…,m then cofactor of an element of A, say $(a_{ij})$, is defined as the product (-1) raised to the power of (i+j) and the Minor of that element. Here, co-factor of element $a_{ij}$ is denoted by $\text{C}_{ij}$.

Mathematically,

$\text{C}_{\text{ij}} \ = \ (-1)^\text{i+j} \times M_{\text{ij}}$

Example: How to find the co-factor of element $a_{11}$ of a matrix A?

Considering the above-mentioned matrix A

$\text{A} = \left ( \displaylines{a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}} \right )_{3\text{x}3}$

STEP 1: Locate the position of the element in the matrix by the values of i and j.

For the element $a_{11}$, it is positioned in the 1^st row and the 1^st column of the matrix A.

STEP 2: Find the minor of the element.

For the element $a_{11}$, its minor is $M_{11} \ = \ (a_{22} \times a_{33} \ – \ a_{23} \times a_{32} )$

STEP 3: Use the mathematical formula for co-factors of elements.

For the element $a_{11}$, cofactor of matrix is $C_{11} \ = \ (-1)^{1+1} M_{11}$

$\therefore \ \text{C}_{11} = \text{M}_{11}$

For the different values of i and j, the co-factor matrix of given matrix A can be written as

$\left ( \displaylines{C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33}} \right )_{3\text{x}3} \ = \ \left ( \displaylines{ M_{11} & – M_{12} & M_{13} \\ – M_{21} & M_{22} & – M_{23} \\ M_{31} & M_{32} & M_{33} } \right )_{3\text{x}3}$

NOTE

Give due priority to the signs involved.