Education
Matrix Calculator
Pick a size (2×2 or 3×3) and an operation. Enter your matrices and the calculator returns the result. Determinant and inverse run on Matrix A alone; addition, subtraction, and multiplication use both A and B.
Matrix size
Operation
Matrix A
Matrix B
This page handles operations between matrices. To solve a system of equations or compute reduced row echelon form, use the system of equations or RREF calculator instead.
Result A × B
0 0 | 0 0
Inverse exists only when det(A) ≠ 0. The 3×3 inverse uses the adjugate method (cofactors transposed, divided by the determinant).
Examples
[[1,2],[3,4]] + [[5,6],[7,8]]
[[6,8],[10,12]]
[[1,2],[3,4]] × [[2,0],[1,2]]
[[4,4],[10,8]]
det([[1,2],[3,4]])
= −2
inv([[2,0],[0,2]])
[[0.5,0],[0,0.5]]
How it works
Matrix operations are well-defined arithmetic on rectangular arrays of numbers. The calculator applies the standard element-wise rules for addition, subtraction, and scalar multiplication, and the row-by-column rule for matrix multiplication.
Multiplication · (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ
2×2 determinant · det = ad − bc
2×2 inverse · A⁻¹ = (1/det)·[[d,−b],[−c,a]]
Related linear algebra calculators
- RREF calculator for reduced row echelon form on 2×2, 2×3, 3×3, or 3×4 matrices.
- System of equations calculator for solving Ax = b directly using Cramer's rule.
- Eigenvalue calculator for the eigenvalues of a 2×2 matrix from trace and determinant.
- Vector calculator for dot product, cross product, magnitude, and angle between vectors.
- Linear regression calculator for the best-fit line through data, which uses matrix arithmetic under the hood.
- All education calculators.
Frequently asked questions
Addition (A + B), subtraction (A − B), scalar multiplication (s · A), matrix multiplication (A × B), transpose (Aᵀ), determinant det(A), and inverse A⁻¹. Both 2×2 and 3×3 matrices are supported.
This page intentionally caps at 3×3 to keep the UI tight and the algorithms numerically reliable. For larger matrices and reduced row echelon form, use the RREF calculator.
When its determinant is nonzero. If det(A) = 0, the matrix is singular and has no inverse. The calculator reports the determinant alongside the inverse so you can see why an inverse may or may not exist.
For 2×2, it uses the closed-form swap-and-negate formula. For 3×3, it uses the adjugate method: take the cofactor matrix, transpose it, and divide every entry by the determinant.
No. To solve a linear system Ax = b, use the system of equations calculator (for 2 or 3 variables) or the RREF calculator (which handles the augmented matrix directly).
For addition, subtraction, and multiplication this calculator assumes A and B are the same square size. Add/subtract requires equal dimensions; multiplication for non-square matrices needs the column count of A to match the row count of B, which is out of scope here.
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