Matrix Calculator
Add, multiply, transpose and invert 2×2 and 3×3 matrices, and compute determinants.
Inputs
Determinant
-2.00000000
Matrix A
1 2 3 4
Singular?
No
Step by step
2×2 matrix A
= 1 2 3 4
det(A)
(1)(4) − (2)(3)
= -2
Singular?
= No
How it works
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
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.