Determinant Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Determinant.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The determinant is a scalar value computed from a square matrix that encodes important geometric and algebraic information. For a 2 \times 2 matrix \begin{bmatrix} a & b \\ c & d \end{bmatrix}, the determinant is ad - bc. A nonzero determinant means the matrix is invertible.

The determinant measures how a matrix scales area (in 2D) or volume (in 3D). If \det(A) = 3, the transformation described by A triples all areas. If \det(A) = 0, the transformation collapses space into a lower dimension (like squishing a plane into a line), which is why the matrix has no inverse.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The determinant tells you whether a matrix is invertible (\det \neq 0) and how the corresponding transformation scales areas or volumes.

Common stuck point: For 3 \times 3 matrices, cofactor expansion can be error-prone. Use the rule of Sarrus or carefully track signs in the checkerboard pattern: +, -, + across the first row.

Sense of Study hint: For a 2x2 matrix, draw an X through it: multiply the main diagonal and subtract the product of the other diagonal.

Worked Examples

Example 1

easy
Find \det\begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}.

Solution

  1. 1
    Step 1: Apply formula: \det = ad - bc where a=3, b=1, c=2, d=4.
  2. 2
    Step 2: \det = 3(4) - 1(2) = 12 - 2 = 10.
  3. 3
    Check: Since \det \neq 0, the matrix is invertible ✓

Answer

10
The 2 \times 2 determinant is computed as ad - bc (product of main diagonal minus product of anti-diagonal). A nonzero determinant means the matrix is invertible.

Example 2

hard
Evaluate \det\begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & 4 \end{bmatrix} by expanding along the first row.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find \det\begin{bmatrix} 5 & 2 \\ 3 & 4 \end{bmatrix}.

Example 2

medium
Is \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} invertible?
← Back to Determinant Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

matrix definitionmatrix multiplication