Systems of Equations and Inequalities: Matrices and Determinants

Determinant of a Matrix

The determinant of a square matrix (one that has the same number of rows as columns) is a real number that can easily tell us whether or not a matrix has an inverse. The inverse will be vital when we use matrices to solve equations in upcoming sections.

Let A = two rows and two columns written inside a set of brackets, row one has a, b, row two has c, d. The determinant of A, denoted det(A) or two rows and two columns written inside two vertical lines, row one has a, b, row two has c, d is defined at det(A) = ad – bc.

Matrix A has an inverse if and only if it is square and det(A) ≠ 0.

For example, given the A = row one is 3, 6 and row two is 5, 2, det(A) = ad – bc = 3(2) – 6(5) = 6 – 30 = -24.

Because this does not equal 0 and matrix A has the same number of rows as columns, matrix A has an inverse.