# Determinant calculator | Determinant of a matrix calculator

## Determinant calculator

Matrix A:

Calculating determinants is one of the most important operations in linear algebra. Therefore, given the importance of this mathematical procedure, we have decided to create the online Determinant Calculator that we offer to you completely free of charge.

To calculate the determinant of a matrix with this calculator, you only need to follow three simple steps:

1. Enter the matrix for which you are going to calculate its slope. To do this, you will use the fields designated for this purpose, being able to increase or decrease the number of columns and rows depending on the size of the matrix that you are going to enter. Remember that to calculate the determinant the matrix must be square.
2. Choose if you want to obtain the solution expressed in decimal numbers or not.
3. Press the “Determinant of the matrix” button. Once you press this button, the solution will be automatically displayed.

The solution is explained step by step using four different methods: the triangle rule, Sarrus’s rule, Cofactor expansion, and Gaussian elimination method.

Remember that the matrix must be square to calculate its determinant.

## What is the determinant of a matrix? | Determinant definition

The concept of determinant can be understood as a function that takes a square matrix as input and returns a number as output. It is important to remember that a square matrix has the same number of rows as it has columns. The determinant of a matrix can also be defined as a scalar property of the matrix. Determinants are very useful in mathematics since they allow us to determine if a matrix is invertible or not, solve systems of simultaneous linear equations using Cramer’s rule, find the area of triangles if the coordinates of their vertices are known, and many other applications. There are several notations to refer to the calculation of determinants, such as the abbreviation “det” followed by the matrix, or the matrix enclosed between two vertical bars, which should not be confused with the absolute value of a matrix. ## Properties of the determinants

1. If all the elements in a row or column are equal to 0, then the value of the determinant is zero.
2. The determinant of an identity matrix (In) is 1 .
3. If rows and columns are swapped, the value of the determinant remains the same. Therefore, det(A) = det(AT) , where AT is the transpose of matrix A.
4. If two rows or two columns of a determinant are swapped, the value of the determinant is multiplied by -1 .
5. If all the elements in a row or column of a determinant are multiplied by some scalar number k, the value of the new determinant is k times the given determinant. Therefore, if A is a square matrix of n rows and K is any scalar. So | KA | = Kn|A| .
6. If two rows (or columns) of a determinant are identical, the value of the determinant is zero.
7. Let A and B be two matrices, then det(AB) = det(A) * det(B) .
8. If A is a matrix, then | An| =(|A|)n .
9. The determinant of the inverse of a matrix can be determined as follows |A-1|=1/|A|
10. The determinant of a diagonal matrix, the triangular matrix (upper triangular or lower triangular matrix) is equal to the product of the elements of the main diagonal.

## How to find the determinant of a matrix

### Determinant of a 2x2 matrix

Given the 2×2 matrix

A =

 a11 a12 a21 a22

We have that its determinant is equal to the product of the elements of the main diagonal minus the product of the elements of the secondary diagonal. ### Determinant of a 3x3 matrix

A =

 a11 a12 a13 a21 a22 a23 a31 a32 a33

To find the determinant of a 3×3 matrix, we first need to perform some multiplication operations. For each element in the first row, we multiply that element by the determinant of the 2×2 matrix that is formed by removing the row and column of that element. This is called the “cofactor method.” Here’s the formula: 