If a matrix has the same number of rows and columns, it is called a square matrix. There is a real number associated with every square matrix, and that number is called the determinant. The determinant of matrix A is written as ½ A½ .

Definition of the Determinant of a 2 x 2 Matrix

Note: Matrices are enclosed in brackets; determinants are denoted with vertical bars.

Example. Evaluate each determinant:

(a) (b)

Solution.

(a) = 5(-3) – 7(6) = -15 – 42 = -57

(b) = 2(-5) – (-3)(4) = -10 + 12 = 2

We can evaluate 3 x 3 determinants using a method called expansion by minors.

The minor of an element of a 3 x 3 determinant is the 2 x 2 determinant that remains after you delete the row and column in which the element appears. For example, in the following determinant, we will find the minor of the element 6:

Since the element 6 appears in the first row and the first column, we delete them.

This leaves the 2 x 2 determinant:

Definition of the Determinant of a 3 x 3 Matrix

When evaluating a 3 x 3 determinant using this definition, keep in mind the following:

1. Each of the three terms is obtained by multiplying an element of the first row by its

minor.

2. A minus sign precedes the second term.

Example. Find the value of the determinant:

Solution.

= 1(-4 - 2) – 2(8 – 6) + (-3)(2 – (-3))

= 1(-6) – 2(2) + (-3)(5)

= -6 – 4 – 15

= -25