How to Use
- Set matrix dimensions
Define the number of rows and columns for your matrix.
- Enter matrix values
Fill in each cell of the matrix. For two-matrix operations, enter the second matrix as well.
- Select operation and calculate
Choose the desired operation and click Calculate to see the result matrix.
What is a Matrix?
A matrix is a rectangular array of numbers, used as a tool to handle systems of equations, linear transformations, and data transformations as a single block. An m×n matrix consists of m rows and n columns, and the value at each position is called an element.
A square matrix, where the number of rows equals the number of columns, is especially important because it can have a determinant and an inverse. This calculator supports six operations: the determinant, inverse, transpose, and scalar multiplication of a single matrix, plus addition and multiplication between two matrices.
Key Applications
- Linear transformations such as rotation and scaling in computer graphics
- Solving systems of linear equations and regression analysis in statistics
- Weight computations in machine learning, and structural analysis in engineering
Calculation Formulas
2×2 determinant: det(A) = ad - bc (A = [[a,b],[c,d]]). Example: for [[1,2],[3,4]], 1×4 - 2×3 = 4 - 6 = -2.
Inverse matrix: A⁻¹ = adj(A) / det(A). For a 2×2 matrix, A⁻¹ = (1/det)·[[d,-b],[-c,a]]. The inverse of the example above is (1/-2)·[[4,-2],[-3,1]] = [[-2,1],[1.5,-0.5]].
Matrix multiplication: C[i][j] = Σ A[i][k]·B[k][j] — the number of columns in A must equal the number of rows in B. Transpose: Aᵀ[i][j] = A[j][i] swaps rows and columns. Determinants of 3×3 and larger matrices are computed recursively using Laplace (cofactor) expansion.