Matrix Calculator
Perform matrix operations: addition, subtraction, multiplication, determinant, inverse and transpose. Supports any rectangular matrix size.
How to use the Matrix Calculator
- Enter your inputs into the Matrix Calculator above.
- Results update instantly as you type — no submit button needed.
- Adjust any value to see how the result changes in real time.
Common matrix operations
AB requires cols(A) = rows(B) · · · det(A) for square A · · · A⁻¹ exists iff det(A) ≠ 0 · · · (Aᵀ)ᵢⱼ = Aⱼᵢ
Matrix multiplication is not commutative (AB ≠ BA generally). Determinant tests invertibility. Transpose swaps rows and columns. Used in linear algebra, computer graphics, statistics, physics.
Worked example
Matrix A = [[1,2],[3,4]]. det(A) = 1×4 − 2×3 = −2 (non-zero, so invertible). A⁻¹ = (1/det) × [[4,−2],[−3,1]] = [[−2,1],[1.5,−0.5]]. Verify: A × A⁻¹ = identity matrix [[1,0],[0,1]].
Frequently asked questions
When is a matrix singular?
When its determinant equals 0. Singular matrices have no inverse and cause systems of linear equations to have either no solution or infinitely many.
What does matrix multiplication represent?
Function composition for linear transformations. Multiplying two matrices represents applying one transformation after another. Used in 3D graphics, robotics and quantum mechanics.
How do I solve a system Ax = b?
If A is invertible: x = A⁻¹b. For non-square or singular A, use Gaussian elimination, LU decomposition or least-squares methods.