# Determinant of a matrix

The determinant of a dimension matrix *mxn** *is the result of subtracting the multiplication of the elements of the main diagonal with the multiplication of the elements of the secondary diagonal.

In other words, the determinant of a 2 × 2 matrix is obtained by drawing an X over its elements. First we draw the diagonal that begins at the top on the left side of the X (main diagonal). Then we draw the diagonal that starts at the top on the right side of the X (secondary diagonal).

To calculate the determinant of a matrix, we need its dimension to have the same number of rows (m) and columns (n). Therefore, *m = n*. The dimension of an array is represented as the multiplication of the row dimension with the column dimension.

There are other more complex ways to calculate the determinant of a matrix with a dimension greater than 2 × 2. These forms are known as Laplace's rule and Sarrus's rule.

The determinant can be indicated in two ways:

- Det (Z)
- | Zmxn |

We call (m) for the dimension of the rows and (n) for the dimension of the columns. So a matrix *m*x*n* will have *m*rows and *n*columns:

*i*represents each of the rows of a matrix Zmxn.*j*represents each of the columns of a matrix Zmxn.

Recommended articles: matrix typologies, inverted matrix.

## Properties of determinants

- | Zmxn | is equal to the determinant of a Zmxn transposed matrix:

- The inverse determinant of an inverse Zmxn matrix is equal to the determinant of an inverse Zmxn matrix:

- The determinant of a singular matrix Smxn (not invertible) is 0.

Smxn = 0

- | Zmxn |, where m = n, multiplied by a constant
*h*any is:

- The determinant of the product of two matrices Zmxny Xmxn, where m = n, is equal to the product of determinants of Zmxny Xmxn

## Practical example

2 × 2 dimension matrix

A dimension array *2×2 *its determinant is the subtraction of the product of the elements of the main diagonal with the product of the elements of the secondary diagonal.

We define Z2 × 2 as:

The calculation of its determinant would be:

### Determiner calculation example

The determinant of the X2 × 2 matrix is 14.

The determinant of the G2 × 2 matrix is 0.

Identity matrix Transposed matrix