Matrix Calculator

Matrix multiplication A × B: Result[i][j] = sum(A[i][k] × B[k][j]) for all k. Requires: columns of A = rows of B. Result size: rows of A × columns of B. Example: 2×3 × 3×2 = 2×2 result. NOT commutative: A×B ≠ B×A in general. Element calculation: for A (2×2) × B (2×2): result[0][0] = A[0][0]×B[0][0] + A[0][1]×B[1][0]. Time complexity: O(n³) for n×n matrices (Strassen: O(n^2.807), rarely used in practice for small matrices).

The determinant (det or |A|) is a scalar value that encodes certain properties of the matrix. For 2×2: det([[a,b],[c,d]]) = ad − bc. For 3×3: cofactor expansion. Properties: det = 0: matrix is singular (not invertible). det ≠ 0: matrix is invertible. det(A×B) = det(A) × det(B). det(Aᵀ) = det(A). det(kA) = kⁿ × det(A) for n×n matrix. Geometric interpretation: |det| = volume of parallelepiped formed by column vectors. Sign determines orientation. Used in: Cramer's rule for solving linear systems, cross products, change of variables in integrals.

Transpose Aᵀ flips the matrix over its main diagonal — rows become columns. (Aᵀ)[i][j] = A[j][i]. Properties: (Aᵀ)ᵀ = A. (A + B)ᵀ = Aᵀ + Bᵀ. (AB)ᵀ = BᵀAᵀ (note reversal). (kA)ᵀ = kAᵀ. Symmetric matrix: A = Aᵀ (e.g., identity matrix). Orthogonal matrix: AᵀA = I (transpose = inverse). Applications: transpose appears in least squares regression (XᵀX)⁻¹Xᵀy, neural network backpropagation, and covariance matrices.

The identity matrix I has 1s on the diagonal and 0s everywhere else. For 3×3: [[1,0,0],[0,1,0],[0,0,1]]. Properties: A × I = I × A = A (multiplicative identity). det(I) = 1. Iᵀ = I. I is the matrix analog of the number 1 in scalar multiplication. Matrix inverse: A⁻¹ is the matrix where A × A⁻¹ = I. Only square matrices with non-zero determinant are invertible. For 2×2: A⁻¹ = (1/det) × [[d,-b],[-c,a]] where A = [[a,b],[c,d]].

For square matrix A: eigenvector v and eigenvalue λ satisfy Av = λv. The eigenvector is a direction that is only scaled (not rotated) by the matrix. Eigenvalues found by: det(A − λI) = 0 (characteristic equation). For 2×2 matrix: this gives a quadratic equation → 2 eigenvalues (possibly complex). Applications: PCA (principal component analysis) in machine learning uses eigenvectors of covariance matrix. PageRank algorithm is an eigenvector problem. Quantum mechanics: observables are eigenvalues of operators. Graph theory: spectral clustering uses eigenvectors of adjacency/Laplacian matrix.