System of Linear Equations

A system of linear equations (SLE) is a set of equations where each equation is a linear combination of variables. The goal is to find a solution that satisfies all the equations simultaneously.

Example

Let's look at the example of a system of linear equations:

We have 3 unknown variables x, y and z and have 3 equations that include all of these variables.

Solving the system

How can we solve the system? Firstly, let's rewrite it in a matrix form:

Expressing the system of linear equations in matrix form provides us with a straightforward approach for solving the system utilizing the inverse matrix:

To use this approach we have to be sure that matrix A can be inversed:
The square matrix A can be inversed if and only if its determinant is nonzero.

Example 1


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import numpy as np

# Define the coefficients matrix A
A = np.array([[2, 3, -1],
              [4, -1, 2],
              [1, 2, -3]])

# Define the constants vector н
y = np.array([10, 4, -1])

# Check the determinant of matrix A
print(f'Determinant is {round(np.linalg.det(A), 3)}')

# Calculating inversed matrix
A_inv = np.linalg.inv(A)
# Finding solution using inversed matrix
x = np.dot(A_inv, y)
    
# Print the solution
print(f'Solution: {np.round(x, 3)}')

We have found the solution using the inverted matrix.

Example 2


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import numpy as np

# Define the coefficients matrix A
A = np.array([[1, 2, 3],
              [3, 1, 4],
              [4, 3, 7]])

# Define the constants vector b
b = np.array([10, 20, 30])

# Check the determinant of matrix A
det_A = np.linalg.det(A)

# Print the determinant value
print(f'Determinant of A: {det_A}')

A_inv = np.linalg.inv(A)

The code above produces an error - the matrix is singular (has zero determinant) so we can't solve the system of equations.
The explanation for this is quite simple: the matrix rows are linearly dependent (the third row is the sum of the first two). As a result, the third equation doesn't provide any additional information, and we are left with a system of 3 variables but only 2 unique equations. As a result such a system either have no solutions or there are lots of solutions.

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Section 2. Chapter 7