Lesson 18.7

Determinants and Cramer's Rule

A single number that reveals whether a matrix is invertible—and a shortcut for solving small systems.

Introduction

The determinant is a scalar value computed from a square matrix that tells us about its properties: if , the matrix is invertible; if , it's singular. For small systems, Cramer's Rule uses determinants to find solutions directly without row reduction.

Prerequisite Connection

You can find matrix inverses and understand when a matrix is invertible.

Today's Increment

We compute determinants of 2×2 and 3×3 matrices, and apply Cramer's Rule.

Why This Matters

Determinants measure area/volume scaling in transformations. Cramer's Rule gives elegant closed-form solutions useful in theoretical physics and engineering.

Key Concepts

2×2 Determinant

Multiply diagonals and subtract.

3×3 Determinant (Expansion)

Expand along the first row using cofactors:

Each cofactor is times the determinant of the submatrix obtained by deleting row and column .

Cramer's Rule

For system with :

where is matrix with column replaced by .

Worked Examples

Example 1: 2×2 Determinant (Basic)

Find the determinant:

Apply formula:

Since , the matrix is invertible.

Example 2: Cramer's Rule 2×2 (Intermediate)

Solve using Cramer's Rule:

Step 1: Find

Step 2: Find (replace column 1 with constants)

Step 3: Find (replace column 2 with constants)

Solution:

and

Example 3: 3×3 Determinant (Advanced)

Find the determinant:

Expand along row 1:

Calculate 2×2 determinants:

Common Pitfalls

Sign errors in cofactor expansion

The pattern is for row 1. Remember alternates signs in a checkerboard pattern.

Using Cramer's Rule when

Cramer's Rule only works for nonsingular matrices. If , use Gaussian elimination instead.

Replacing the wrong column

For , replace column of with the constants vector. Double-check which variable you're solving for.

Real-World Application

Area and Volume Calculations

The absolute value of a 2×2 determinant gives the area of the parallelogram spanned by two vectors. For 3×3, it gives the volume of the parallelepiped. This is why transformations that have determinant 2 double areas, and determinant 0 collapse space to lower dimensions.

Area = for vectors and