Page 2 - Lesson Notes

Let A be a matrix of order n and let |A| = x. Then, |kA| = k |A| = k x,
n
n
where n = 1, 2, 3,…

Minor: Minor of an element ay of a determinant, is a determinant obtained by
deleting the ith row and jth column in which element ay lies. Minor of an
element a ij is denoted by M ij.
Note: Minor of an element of a determinant of order n(n ≥ 2) is a determinant
of order (n – 1).

Cofactor: Cofactor of an element a ij of a determinant, denoted by A ij or C ij is
defined as = (-1) M ij, where M ij is a minor of an element a ij.
i+j

Note
(i) For expanding the determinant, we can use minors and cofactors as
| A | = a M – a M + a M = a A + a A + a A
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(ii) If elements of a row (or column) are multiplied with cofactors of any other
row (or column), then their sum is zero.
Singular and non-singular Matrix: If the value of determinant corresponding
to a square matrix is zero, then the matrix is said to be a singular matrix,
otherwise it is non-singular matrix, i.e. for a square matrix A, if |A| ≠ 0, then it
is said to be a non-singular matrix and of |A| = 0, then it is said to be a
singular matrix.
Theorems
(i) If A and B are non-singular matrices of the same order, then AB and BA are
also non-singular matrices of the same order.
(ii) The determinant of the product of matrices is equal to the product of their
respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix
of the same order.

Adjoint of a Matrix: The adjoint of a square matrix ‘A’ is the transpose of the
matrix which obtained by cofactors of each element of a determinant
corresponding to that given matrix. It is denoted by adj(A).
In general, adjoint of a matrix A = [a ij] n×n is a matrix [A ji] n×n, where A ji is a cofactor
of element a ji.

Properties of Adjoint of a Matrix
If A is a square matrix of order n × n, then

 A(adj A) = (adj A)A = |A| I n