To define the determinant of a square matrix of order 3 or higher, it is convenient to introduce the concepts of minors and co-factors.
The minor of an entry in a n × n matrix is the determinant of the (n − 1) × (n− 1) matrix that remains after deletion of the row and column in which the entry occurs.
If A is a square matrix of order n, then the minor of an entry aij is the determinant of the matrix that remains after deletion of the ith row and jth column of A. It denoted by Mij.
The co-factor of an entry aij is the product of (– 1)i + j and the minor of aij. It denoted by Cij.
i.e, Cij = (−1)i + j Mij
i. The co-factor of an element is equal to its minor if (i + j) is even.
ii. The co-factor of an element is equal to the negative of its minor if (i + j) is odd.
Note: Minors and co-factors of remaining entries are obtained in similar manner.
For any square matrix of order n, the determinant of matrix is the sum of the entries in any row (or column) of a matrix multiplied by their respective co-factors.
For example, if be a square matrix of dimension 3,
then the determinant of A is given as:
|A| = a11(co-factor of a11) + a12(co-factor of a12) + a13(co-factor of a13)
Applying this definition to find a determinant is called expanding by co-factors along the first row.
|A| = a11(–1)1+1. M11 + a12(–1)1+2. M12 + a13(–1)1+3. M13
= a11M11 – a12M12 + a13M13
The same result for the value of the determinant is obtained by expanding by co-factors along other rows or columns.
This is explained under
The above process can be extended to square matrices of order 4 or more.
The formulae for determinant of an n × n square matrix are given under