Chapter 6: The determinant

What is a Matrix Determinant?

This special scaling factor, the factor by which a linear transformation changes areas, is called the “determinant” of that transformation.

A matrix determinant is a special number that can be calculated from a square matrix (a matrix with the same number of rows and columns). Think of the determinant as a summary of how the matrix transforms space. It gives important information about the properties of the matrix, especially when working with linear transformations, such as rotations, scaling, or shearing.

The determinant is only defined for square matrices (matrices with the same number of rows and columns). If a matrix is not square, it doesn't have a determinant.

This is because the determinant is a mathematical property of linear transformations represented by square matrices, where it gives information about the volume scaling factor and whether the matrix is invertible. Non-square matrices don't represent such transformations, so the concept of a determinant doesn't apply to them.

In general, larger matrices (3x3, 4x4, etc.) have more complex methods for finding the determinant, but the concept remains the same—it represents something essential about the matrix and how it transforms objects in space.

import numpy as np

# Define a square matrix
matrix = np.array([[1, 2], [3, 4]])

# Calculate the determinant
det = np.linalg.det(matrix)

# Print the result
print(f"Determinant: {det}")

What is the Use of a Matrix Determinant?

1. Understanding Transformations: The determinant helps you understand how a matrix changes or transforms things. For example, if the determinant is zero, it means the matrix squashes the space into a lower dimension, and information is lost. This tells us that the transformation can't be reversed (it's not invertible).
2. Volume Scaling: The determinant shows how much a matrix stretches or shrinks space. If the determinant is 1, the matrix preserves the space's size (it doesn’t stretch or shrink it). If the determinant is 2, it doubles the size of the space, and if it’s 0.5, it shrinks the space by half. A negative determinant means the matrix flips or reflects the space as well as stretches or shrinks it.
3. Checking Invertibility: A matrix is invertible (meaning you can reverse its transformation) if its determinant is non-zero. If the determinant is zero, the matrix has lost some information (such as squashing space into a line), and it can't be undone.

One of the most insightful ways to understand linear transformations and matrices is by examining how they scale areas in space. This is closely related to the determinant of a matrix, a concept that tells you how much a transformation stretches, compresses, or even squashes areas or volumes. Let’s explore this idea step by step with a detailed explanation and examples.