## 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)

$\matrix(2, 4 ; -3, 1)$

## 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.

## Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva