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Matrix Calculator — Addition, Multiplication, Inverses & Determinants
Perform linear algebra matrix operations online: addition, subtraction, matrix multiplication, determinants, adjugate inverses, transpositions, and trace for 2×2 up to 5×5 matrices.
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Published
Select Matrix Operation
Matrix A (2 × 2)
Size:
Matrix B (2 × 2)
Size:
Result — Multiplication (A × B)
Output Matrix (2 × 2)
4
7
2
6
MMaintained by Mathematics Tools Desk
Published
Calculation Methodology & Formula
Computes matrix multiplication C = A × B where c_ij = Σ a_ik * b_kj. Determinants computed via recursive Laplace cofactor expansion. Inverse computed via adj(A) / det(A).
C_{i,j} = \sum_{k=1}^m A_{i,k} \cdot B_{k,j} \quad \Big| \quad A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \quad \Big| \quad \text{tr}(A) = \sum_{i=1}^n A_{i,i}Worked Example: 2×2 Matrix Inversion
Find the determinant and inverse of matrix [[4, 7], [2, 6]].
matrix A:[[4, 7], [2, 6]]
operation:Matrix Inverse (A⁻¹)
Expected Outputdet(A) = 10, A⁻¹ = [[0.6, -0.7], [-0.2, 0.4]]
det = (4)(6) - (7)(2) = 10. Inversion multiplies adjugate by 1/10.
Technical Constraints & Limitations
- Matrix inverse requires a square matrix with a non-zero determinant (det ≠ 0).
- Matrix multiplication requires the number of columns in A to equal the number of rows in B.
Frequently Asked Questions
When can two matrices be multiplied together?
Matrix multiplication A × B is valid only when the number of columns in Matrix A equals the number of rows in Matrix B. The resulting product matrix has dimensions (rows of A) × (columns of B).
What is a singular matrix?
A singular matrix is a square matrix whose determinant is zero (det = 0). A singular matrix has no multiplicative inverse.
What is the Trace of a matrix?
The trace of a square matrix is the sum of its main diagonal elements (from top-left to bottom-right).