Determinant Calculator
Calculate matrix determinant det(A) for 2x2 and 3x3 square matrices step-by-step using cross product and Laplace expansion.
Matrix Determinant Engine
Select matrix size and enter matrix elements.
Determinant Output
Quick Summary
Our determinant calculator 2x2 3x3 matrix steps tool computes the determinant scalar value $\det(A)$ for square matrices. It uses cross product ($ad - bc$) for 2×2 matrices and Laplace cofactor expansion across Row 1 (or Sarrus Rule) for 3×3 matrices, testing matrix singularity and geometric scaling.
How It Works: Determinant Formulas & Cofactor Expansion
The determinant measures linear transformation scaling and invertibility:
1. **2×2 Determinant Formula:** $\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc$.
2. **3×3 Cofactor Expansion:** $\det(A) = a_{11} M_{11} - a_{12} M_{12} + a_{13} M_{13}$ where $M_{ij}$ are 2×2 minor determinants.
3. **Rule of Sarrus (3×3 shortcut):** Sum of 3 downward diagonal products minus sum of 3 upward diagonal products.
4. **Invertibility Criterion:** $\det(A) \neq 0 \iff A$ is non-singular and invertible ($A^{-1}$ exists).
5. **Geometric Area / Volume:** $|\det(A)|$ gives area expansion factor in 2D (parallelogram area) or volume expansion factor in 3D (parallelepiped volume).
Formula Explanation
Your determinant calculations follow classical Laplace expansion formulas:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for calculating the determinant of 2×2 matrix $A = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$:
- Step 1 (Identify Elements): $a_{11} = 4, a_{12} = 7, a_{21} = 2, a_{22} = 6$.
- Step 2 (Primary Diagonal Product): $a_{11} \times a_{22} = 4 \times 6 = \mathbf{24}$.
- Step 3 (Secondary Diagonal Product): $a_{12} \times a_{21} = 7 \times 2 = \mathbf{14}$.
- Step 4 (Subtract Products): $\det(A) = 24 - 14 = \mathbf{10}$.
- Step 5 (Check Invertibility & Geometry): $\det(A) = 10 \neq 0 \implies$ Non-singular invertible matrix with area scaling factor 10, delivering **det(A) = 10**!
Calculation Examples: Real-World Scenario Comparison
Compare matrix dimensions, inputs, computed determinants, and singularity status:
| Matrix Inputs | Dimension | Expansion Calculation | det(A) | Invertibility Status |
|---|---|---|---|---|
| [[4, 7], [2, 6]] | 2 × 2 | (4 × 6) - (7 × 2) | 10 | Non-Singular (Invertible) |
| [[2, 4], [1, 2]] | 2 × 2 | (2 × 2) - (4 × 1) | 0 | Singular Matrix (No A⁻¹) |
| [[1, 0, 0], [0, 1, 0], [0, 0, 1]] | 3 × 3 | 1(1-0) - 0 + 0 | 1 | Identity Matrix (Invertible) |
| [[1, 2, 3], [4, 5, 6], [7, 8, 9]] | 3 × 3 | 1(-3) - 2(-6) + 3(-3) | 0 | Singular Matrix (Collinear) |
Benefits of Using the Determinant Calculator
Utilizing this calculator provides essential linear system solving (Cramer's Rule), eigenvalues, and vector calculus advantages:
- Cramer's Rule Linear System Solver: Computes system determinants ($D, D_x, D_y, D_z$) to solve linear systems.
- Singularity & Rank Test: Instantly verifies if a matrix is non-singular ($\det(A) \neq 0$) before computing inverse $A^{-1}$.
- Eigenvalue Characteristic Polynomial ($\det(A - \lambda I) = 0$): Evaluates scalar shifts for matrix diagonalizations.
- Vector Cross Product & Triple Product: Computes 3D vector cross products and parallelepiped volumes.
Frequently Asked Questions (FAQ)
What is a matrix determinant?
The determinant is a scalar value calculated from a square matrix that encodes linear transformation scaling, invertibility, and volume scaling.
Can you calculate determinant for non-square matrices (e.g. 2x3)?
NO. Determinants are defined ONLY for square matrices (n × n).
What happens if det(A) = 0?
If det(A) = 0, the matrix is singular, its rows are linearly dependent, and it has NO matrix inverse (A^-1 does not exist).
What is Sarrus' Rule?
Sarrus' Rule is a visual shortcut for 3x3 determinants: add products of 3 main diagonals and subtract products of 3 anti-diagonals.
What is the determinant of a triangular matrix?
For an upper or lower triangular matrix, the determinant equals the product of its main diagonal elements.
What is the determinant of the transpose det(A^T)?
det(A^T) = det(A).
What is the determinant of an inverse det(A^-1)?
det(A^-1) = 1 / det(A).
What is det(cA) for an n x n matrix?
det(cA) = c^n × det(A) (e.g. for 2x2, det(2A) = 4 det(A)).
What happens to det(A) if two rows are swapped?
Swapping any two rows or columns negates the sign of the determinant: det(A') = -det(A).
Why is [[1, 2, 3], [4, 5, 6], [7, 8, 9]] determinant 0?
Because the second row minus first row equals third row minus second row (linearly dependent rows ⇒ det = 0).