Matrix Calculator
Add, subtract and multiply 2×2 or 3×3 matrices, and find the determinant and inverse.
Results
Quick Answer
To multiply matrices, multiply each row of A by each column of B and add the products. For A = [1 2; 3 4] and B = [5 6; 7 8], A × B = [19 22; 43 50]. The determinant of A is 1 × 4 − 2 × 3 = −2.
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Table of Contents
How This Matrix Calculator Works
A matrix is a grid of numbers. Choose a size and an operation, enter your values, and press Calculate. Addition and subtraction work cell by cell, while multiplication combines rows with columns.
Quick explanation: each result cell in A × B is a row of A multiplied by a column of B.
- Size: 2 × 2 or 3 × 3.
- Determinant: a single number that describes the matrix.
- Inverse: exists only if the determinant is not 0.
Core Formulas
- 2 × 2 determinant: ad − bc
- 2 × 2 inverse: (1 ÷ (ad − bc)) × [d −b; −c a]
- Product: cell (i, j) = the sum of A(i, k) × B(k, j)
Worked Example
Find the inverse of A = [1 2; 3 4].
- Determinant = (1 × 4) − (2 × 3) = −2
- Swap and change signs: [4 −2; −3 1]
- Divide by −2: [−2 1; 1.5 −0.5]
Common Mistakes
- Order matters: A × B is usually not the same as B × A.
- Multiplying cell by cell: matrix multiplication uses rows and columns.
- Zero determinant: the matrix has no inverse.
Frequently Asked Questions
Multiply each row of the first matrix by each column of the second, adding the products in each case.
For a 2 × 2 matrix, it is ad − bc. For a 3 × 3 matrix, this calculator expands along the first row.
When its determinant is 0.
No. A × B and B × A usually give different results.
It supports 2 × 2 and 3 × 3 square matrices.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Matrix (mathematics) | Matrix operations. | https://en.wikipedia.org/wiki/Matrix_(mathematics) |
| Wikipedia, Determinant | Determinant formulas. | https://en.wikipedia.org/wiki/Determinant |
| Wikipedia, Invertible matrix | Matrix inverse. | https://en.wikipedia.org/wiki/Invertible_matrix |