Determinant of a square matrix

Introduction:

Every square matrix can be associated with a real number called determinant.

The determinant of a matrix A is written as |A| or det A.

Finding the determinant depends on the dimension of the matrix A.

Determinants exist only for square matrices.

Determinant of a 2 X2 matrix

A= , |A|=det A=det =ad-bc,(multiply the leading diagonal)

Note: the vertical bars, when used in the context of matrices, represents determinants and not absolute value.

Example: find the determinant of the matrix

Let A=

det( A) = |A|= (-6x 1) – (2x7)=-6-14=-20

Rule:

• The determinant of a matrix of order 1x1 is defined as the entry of the matrix.
  Example: If A = [3], then |A| = 3
• The determinant of a 2x2 matrix can be found by the method above.
• The determinant of a 3x3, 4x4 etc. matrix can be obtained by using minors and cofactors.

Minors and Cofactors:

Definition:

If A is a square matrix, the minor M_ij of the entry a_ij is the determinant of the matrix obtained by deleting the i^th row and the j^th column of A.

In other words, the minor of an element is the value of the determinant found by deleting the row and column of that element.

Example: Find the minor of 1 of the matrix C =

The minor of 1 is

= 15 - 8 = 7

Definition: If A is a square matrix, with minor M_ij of the entry a_ij , then the cofactor C_ij of the entry a_ij is

C_ij = (-1)^i+j M_ij

Example: In the above example the cofactor of 1 is C₁₁ = (-1)¹⁺¹ M₁₁ =(-1)² 7 = 7

Sign pattern for Cofactors:

The + and – signs represent (-1)^i+j  for each element.

Using Cofactors to find the determinant of a square matrix:

|A| = a₁₁C₁₁ + a₁₂C₁₂ +… + a_1nC_1n

Example: Find the determinant by expanding along first row of matrix A =

The minor of 1 is 7 and the cofactor is 7.

The minor of 0 is = -51 and the cofactor is 51.

The minor of -3 is = -52 and the cofactor is -52.

Then det (A) = 1(7) + 0(51) +(-3) (-52) = 163.

Properties of Determinants:

• The value of a determinant is unaltered by interchanging its rows and columns.
• If any two rows ( or columns) of a determinant are interchanged the determinant changes its sign but its numerical value unaltered.
• If two rows ( or columns) of a determinant are identical then the value of the determinant is 0.
• If every element in a row ( or column) of a determinant is multiplied by a constant “k” then the value of the determinant is multiplied by k.
• If element in any row ( column) can be expressed as the sum of two quantities then given determinant can be expressed as the sum of two determinants of the same order with the elements of the remaining rows (column)  of both being the same.
• A determinant is unaltered when to each element of any row ( column) is added to those of several other rows(columns) multiplied respectively by constant factors.