Linear Algebra 103

Building on our understanding of 2-variable systems from Linear Algebra 102, we now extend to 3-variable systems and introduce the powerful numpy.linalg sub-library—your toolkit for computational linear algebra.

System of Linear Equations with 3 Variables

Consider this system of linear equations with three equations and three unknowns:

To solve this system means to find values of $x_1$, $x_2$, $x_3$ that satisfy all three equations simultaneously.

Solving with NumPy

First, let’s import NumPy and represent our system as a matrix $A$ and vector $b$:

import numpy as np

A = np.array([
        [4, -3, 1],
        [2, 1, 3],
        [-1, 2, -5]
    ], dtype=np.dtype(float))

b = np.array([-10, 0, 17], dtype=np.dtype(float))

print("Matrix A:")
print(A)
print("\nArray b:")
print(b)

Output:

Matrix A:
[[ 4. -3.  1.]
 [ 2.  1.  3.]
 [-1.  2. -5.]]

Array b:
[-10.   0.  17.]

Let’s verify the dimensions:

print(f"Shape of A: {np.shape(A)}")
print(f"Shape of b: {np.shape(b)}")

Output:

Shape of A: (3, 3)
Shape of b: (3,)

Now we can solve the system using np.linalg.solve(A, b):

x = np.linalg.solve(A, b)

print(f"Solution: {x}")

Output:

Solution: [ 1.  4. -2.]

The solution is $x_1 = 1$, $x_2 = 4$, $x_3 = -2$. You can verify by substituting these values back into the original equations!

Evaluating the Determinant

Matrix $A$ is a square matrix (same number of rows and columns). For square matrices, we can calculate the determinant—a scalar that characterizes important matrix properties.

Key insight: A linear system has one unique solution if and only if the matrix has a non-zero determinant.

d = np.linalg.det(A)

print(f"Determinant of matrix A: {d:.2f}")

Output:

Determinant of matrix A: -60.00

Since the determinant is non-zero ($-60$), our system has exactly one solution—as we found!

What Happens with No Unique Solution?

Let’s explore what happens when a system has no unique solution (either no solution or infinitely many).

Consider this system:

A_2= np.array([
        [1, 1, 1],
        [0, 1, -3],
        [2, 1, 5]
    ], dtype=np.dtype(float))

b_2 = np.array([2, 1, 0], dtype=np.dtype(float))

print(np.linalg.solve(A_2, b_2))

This raises an error:

LinAlgError: Singular matrix

NumPy throws a LinAlgError because the matrix is singular. Let’s check the determinant:

d_2 = np.linalg.det(A_2)

print(f"Determinant of matrix A_2: {d_2:.2f}")

Output:

Determinant of matrix A_2: 0.00

The determinant is zero! This confirms the matrix is singular—it cannot have a unique solution.

Understanding Singular Matrices

When a matrix is singular (determinant = 0), the system either:

• Has no solution (inconsistent system—like parallel lines that never meet)
• Has infinitely many solutions (like identical planes in 3D space)

The np.linalg sub-library contains many powerful linear algebra functions. As you learn more theory, these functions will become clearer and more useful.

Important: Always check for singular matrices! np.linalg.solve() will error if there are no or infinitely many solutions. When using it in production code, wrap it in error handling to prevent crashes.

Summary

In this post, we’ve explored:

1. Extension to 3 variables: The same matrix concepts from 2D extend naturally to higher dimensions
2. The numpy.linalg sub-library: Your computational toolkit for linear algebra
3. Determinants indicate solution type:
   • Non-zero determinant → unique solution
   • Zero determinant → no unique solution (singular matrix)
4. Error handling: np.linalg.solve() throws errors for singular systems

While using np.linalg.solve() is convenient, it gives little insight into what happens “under the hood.” Next, we’ll explore Gaussian Elimination—a fundamental method to solve linear systems that reveals the mechanics behind matrix solutions.

These tools form the computational backbone of machine learning, computer graphics, optimization, and scientific computing.