How to calculate the determinant of a matrix or ndArray using numpy in Python?

In this article, we will learn how to calculate the determinant of a matrix using the numpy library in Python. The determinant of a matrix is ​​a scalar value that can represent the matrix in compact form. It is a useful quantity in linear algebra and has numerous applications in various fields including physics, engineering, and computer science.

In this article, we will first discuss the definition and properties of determinants. We will then learn how to use numpy to calculate the determinant of a matrix and see how it is used in practice through some examples.

Definition and properties of determinant

The determinant of a matrix is ​​a scalar value that can be used to describe the properties of a matrix in a compact form. It is often denoted by either |A| or det(A), where A is the matrix. determinant is a fundamental concept in linear algebra and has several important properties that make it a powerful tool in mathematical calculations.

• One of the most striking properties of the determinant is that it is equal to the product of the eigenvalues ​​of the matrix. Eigenvalues ​​are a special set of scalar values ​​that represent how a matrix acts on certain vectors, and play a crucial role in many applications of linear algebra.

• Another important property of the determinant is that it is equal to the product of the diagonal elements of an upper triangular matrix or a lower triangular matrix. A triangular matrix is ​​a matrix with zeros above or below the diagonal. This property is very useful when calculating the determinant of a large matrix.

• The determinant can also be calculated by multiplying the sum of the elements in any row or column with the appropriate sign. This property provides an alternative method of computing the determinant and is helpful when the matrix is ​​not triangular.

• In addition, the determinant can be calculated by multiplying the elements on the main diagonal of the matrix and dividing by the determinant of the cofactor, submatrix, or adjoint matrix. These matrices are derived from the original matrices and have unique properties that help calculate the determinant.

Use numpy to calculate the determinant of a matrix

Using numpy to calculate the determinant of a matrix, we can use the linalg.det() function. This function accepts a matrix as input and returns the determinant of the matrix. Let’s see an example −

import numpy as np
# create a 2x2 matrix
matrix = np.array([[5, 6], [7, 8]])
# calculate the determinant of the matrix
determinant = np.linalg.det(matrix)
print(determinant)

Output

<font face="Liberation Mono, Consolas, Menlo, Courier, monospace"><span style="font-size: 14px;">-2.000000000000005</span></font>

Code explanation

As you can see, the linalg.det() function calculates the determinant of a matrix and returns it as a scalar value. In this case, the determinant of the matrix is ​​-2.0.

Calculate the determinant of a high-dimensional matrix

To calculate the determinant of a high-dimensional matrix, we can use the same linalg.det() function. Let’s see an example −

import numpy as np
# create a 3x3 singular matrix
matrix = np.array([[20, 21, 22], [23, 24, 25], [26, 27, 28]])
# calculate the determinant of the matrix
determinant = np.linalg.det(matrix)
print(determinant)

Output

2.131628207280298e-14

Code explanation

As you can see, the linalg.det() function can also be used to calculate the determinant of high-dimensional matrices. In this case, the determinant of the matrix is ​​0.0.

Calculate the determinant of a singular matrix

A singular matrix is ​​a matrix without an inverse matrix. The determinant of a singular matrix is ​​0, which means it is not invertible. Let’s look at an example −

The Chinese translation of

Example 1

is:

Example 1

In the following example, the linalg.det() function returns 0 for a singular matrix, which means it is not invertible.

import numpy as np
# create a 3x3 matrix
matrix = np.array([[10, 11, 12], [13, 14, 15], [16, 17, 18]])
# calculate the determinant of the matrix
determinant = np.linalg.det(matrix)
print(determinant)

Output

0.0

The Chinese translation of

Example 2

is:

Example 2

linalg.slogdet() function returns the sign of the matrix and the logarithm of the determinant. The determinant is calculated using the LU decomposition method, which is more stable and accurate than the method used by the linalg.det() function.

One advantage of using the linalg.slogdet() function is that it is more stable and accurate than the linalg.det() function, especially for large matrices. However, remember that it returns the logarithm of the determinant, so you need to exponentiate the result to get the actual determinant

import numpy as np
# create a 3x3 matrix
matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
# calculate the determinant of the matrix using the linalg.slogdet() function
sign, determinant = np.linalg.slogdet(matrix)
print(determinant)

Output

-inf

in conclusion

This article teaches us how to use Python numpy to calculate the determinant of a matrix. We looked at the definition and properties of determinants, and how to use the linalg.det() function to calculate the determinant of a matrix. We also looked at some examples to see how it works in practice. We also learned how to calculate the determinant of a matrix using numpy in Python.

The determinant is a scalar value that can be used to represent a matrix in a concise form. It has many applications in various fields. To calculate the determinant of a matrix using numpy, we can use the linalg.det() function, which accepts a matrix as input and returns the determinant. Alternatively, we can use the linalg.slogdet() function, which returns the sign and logarithm of the determinant using the LU decomposition method. Both functions make it easy to calculate the determinant of a matrix in Python, and they are useful tools for working with matrices in scientific and engineering applications.

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