## Contents

## THE DETERMINANT OF A MATRIX

## Definition

Definition: Let A = [aij] be an n x n matrix. The **determinant** function, denoted by **det**, is defined by

det(A) = $ \sum{a1j1 a2j2... anjn} $

where the summation is over all permutations j1, j2... jn of the set S = {1, 2, ..., n}. The sign is taken as + or - according to whether the permutation j1, j2, ... jn is even or odd.

**2 x 2 Matrices**

**3 x 3 Matrices**

**4 x 4 Matrices**

## Example (2 x 2 Matrix)

A = [4, 2 ; -3 , 1]

det(A) = (4)*(1) - (2)*(-3) = 10

## Example (3 x 3 Matrix)

A = [2, 5, 1 ; 3, 2, 2 ; 1, 5, 2]

det(A) = 2*[(2)*(2) - (2)(5)] - 5*[(3)*(2) - (2)*(1)] + 1*[(3)*(5) - (2)*(1)] = -19

## Example (4 x 4 Matrix)

## Important Properties of the Determinant

- The determinant of the transpose of a matrix will be the same as the determinant of the original matrix.

- If two rows in the matrix are interchanged, the determinant will be the negative of the determinant of the original matrix.

- If two rows or columns in a matrix are the same, the determinant will be zero.

- If there is a row of zeros in the matrix, the determinant will be zero.

- If a row or column in a matrix was multiplied by a real number, then the determinant will be that real number multiplied by the determinant of the original matrix.

- If the matrix is either an upper or a lower triangular matrix, the determinant will be the product of the elements on the main diagonal.

- If two matrices are n x n, then the determinant of one multiplied by the determinant of the other is the determinant of the full matrix of the two next to each other.

## Nonsingular and Similar Matrices

Definition: A square matrix is called **nonsingular** if its determinant is not equal to zero.

- For square matrices, then the linear combination of the matrix has a nontrivial solution if and only if the matrix is NOT nonsingular.

- For nonsingular matrices, the determinant of the inverse of that matrix is equal to 1 over the determinant of the matrix.

- If two matrices are similar matrices, then the determinants of the two are equal.