Matrix Algebra Tools & References

Step-by-step guides and reference pages for core matrix operations. Each page covers the theory, formulas, worked examples, and real-world applications.

Available Tools

Matrix Multiplication Calculator

Multiply matrices with step-by-step dot product breakdowns. Covers dimension compatibility rules (m×n × n×p = m×p), computational complexity O(n³), and applications in neural networks, computer graphics, and graph theory.

Matrix Determinant Calculator

Calculate determinants for 2×2, 3×3, and larger matrices using cofactor expansion. Understand the geometric interpretation (area and volume scaling), properties like det(AB) = det(A)·det(B), and how a zero determinant indicates a singular matrix.

Inverse Matrix Calculator

Find the inverse of a matrix using the Gauss-Jordan elimination method. Includes the 2×2 shortcut formula, the augmented matrix approach, invertibility conditions (det ≠ 0), and applications in solving linear systems and the normal equation for linear regression.

Eigenvalue & Eigenvector Calculator

Compute eigenvalues and eigenvectors via the characteristic polynomial det(A - λI) = 0. Covers diagonalization, the spectral theorem for symmetric matrices, and applications in PCA (Principal Component Analysis), dynamical systems, and PageRank.

Matrix Transpose Calculator

Transpose any matrix by swapping rows and columns. Learn the product rule (AB)^T = B^TA^T, symmetric matrices (A = A^T), orthogonal matrices (Q^T = Q^-1), and how the transpose drives gradient computation in backpropagation.

Try the Interactive Calculator

All of these operations are available in the ML3X Matrix Calculator, which supports matrices up to 5×5 with full step-by-step solution display. Enter your matrices, select an operation, and see every intermediate calculation. No signup required — everything runs in your browser.

Built by Michael Lip. Try the ML3X Matrix Calculator for interactive step-by-step solutions.