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 mxn will have mrows and ncolumns:

• irepresents each of the rows of a matrix Zmxn.
• jrepresents each of the columns of a matrix Zmxn.

Recommended articles: matrix typologies, inverted matrix.

Properties of determinants

1. | 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

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