Matrix Calculator
Add, subtract, multiply, transpose, and find the determinant of 2×2 and 3×3 matrices.
Try an example
[4, 7] [2, 6]
About this calculator
How This Matrix Calculator Works
Pick a matrix size (2×2 or 3×3) and an operation, fill in the values, and the result updates live — addition, subtraction, multiplication, transpose, scalar multiplication, and the determinant.
Worked Example
Multiplying A = [[4,7],[2,6]] by the identity matrix returns A unchanged: [[4,7],[2,6]]. The determinant of A is 4×6 − 7×2 = 10.
The Operations
- Addition/Subtraction — combine matrices of the same size element by element.
- Multiplication — each entry in the result is the dot product of a row from A and a column from B; unlike ordinary multiplication, order matters (A × B usually isn't the same as B × A).
- Transpose — flips a matrix over its diagonal, turning rows into columns.
- Scalar multiplication — multiplies every entry by a single number k.
- Determinant — a single number computed from a square matrix's entries, using cross-multiplication for 2×2 (ad − bc) or cofactor expansion for 3×3.
Why the Determinant Matters
A determinant of zero means the matrix is singular — it can't be inverted, and the system of linear equations it represents either has no solution or infinitely many, rather than exactly one. A nonzero determinant guarantees a unique solution exists. Determinants also show up in geometry as area (2×2) or volume (3×3) scaling factors.
Common Uses
Matrices are the standard way to represent and solve systems of linear equations, describe geometric transformations (rotation, scaling, reflection) in graphics and game development, and organize data in statistics and machine learning — a spreadsheet of numbers is, mathematically, just a matrix.
Worked example
Multiplying A = [[4,7],[2,6]] by the identity matrix B = [[1,0],[0,1]] returns A unchanged — [[4,7],[2,6]]. The determinant of A is 4×6 − 7×2 = 10.
Frequently asked questions
What sizes of matrices does this support?
2×2 and 3×3 — the two sizes covered in most introductory linear algebra. Addition, subtraction, and multiplication require both matrices to be the same size.
Why does A × B not always equal B × A?
Matrix multiplication isn't commutative like regular number multiplication — the order matters, and swapping A and B generally produces a completely different result (when it's even defined, for non-square cases).
What does the determinant tell you?
It's a single number that captures key properties of a matrix — a determinant of zero means the matrix is 'singular' (not invertible, and the system of equations it represents doesn't have a unique solution). It's also used to compute area/volume scaling factors in geometry.
What is a transpose used for?
Flipping a matrix over its diagonal (rows become columns) shows up constantly in linear algebra — from computing dot products to reorienting data in machine learning and statistics.