Matrix Calculator
Add, multiply, transpose and invert matrices, and find determinants.
About Matrix Calculator
A matrix is a rectangular array of numbers. The determinant of a square matrix is a scalar that encodes many properties: a matrix is invertible if and only if its determinant is non-zero. The inverse A⁻¹ satisfies A × A⁻¹ = I (the identity matrix). Transposing swaps rows and columns. Matrix multiplication is not commutative: A × B ≠ B × A in general.
Formulas
2×2 determinant
det [[a,b],[c,d]] = ad − bc
2×2 inverse
[[a,b],[c,d]]⁻¹ = (1/(ad−bc)) × [[d,−b],[−c,a]]
Matrix multiplication element
(AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ
Worked example
With the default values below, Matrix Calculator returns Determinant of -2.00000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Operationone of Determinant (single matrix), Inverse (single matrix), Transpose (single matrix), Add (A + B), Multiply (A × B)
- Determinant (single matrix)
- Matrix A (rows separated by newlines, values by spaces)
- 1 2 3 4
- Matrix B
- 5 6 7 8
Results
- Determinant
- -2.00000000
- Matrix A
- 1 2 3 4
- Singular?
- No
Step by step
- 12×2 matrix A= 1 2 3 4
- 2det(A)= -2
- 3Singular?= No
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is the determinant of [[1,2],[3,4]]?
det = (1)(4) − (2)(3) = 4 − 6 = −2. Since it's non-zero, the matrix is invertible.
What does 'singular' mean for a matrix?
A singular matrix has determinant zero and cannot be inverted. It means the rows (or columns) are linearly dependent — one row can be expressed as a combination of the others. Geometrically it means the transformation collapses space onto a lower dimension.
Can I multiply a 2×2 by a 3×3 matrix?
No. Matrix multiplication A × B requires the number of columns in A to equal the number of rows in B. A 2×2 times a 3×3 has dimension mismatch. This calculator handles square 2×2 and 3×3 matrices only.