Matrix Engine

Mathematical Principles & Theorems

Studies linear transformations on finite-dimensional vector spaces \(T: \mathbb{R}^n \to \mathbb{R}^m\). Matrix multiplication \((AB)_{ij} = \sum_{k} A_{ik} B_{kj}\). Determinant \(\det(A)\) gives signed hypervolume scaling; \(\det(A) \ne 0 \iff A\) is invertible. Inverse \(A^{-1} = \frac{1}{\det(A)}\text{adj}(A)\). Gaussian elimination with partial pivoting reduces augmented matrix \([A|I]\) to Reduced Row Echelon Form (RREF), determining rank, nullity, and linear independence.

Operating Instructions

• Set matrix dimensions (rows \(m\), columns \(n\)) and enter numeric cell values.
• Select matrix operation: Addition, Multiplication, Determinant, Transpose, Inverse, Rank, or RREF.
• Click Calculate to view exact results and step-by-step row operation elementary matrices.
• Inspect geometric interpretation readouts for linear transformation scaling and subspace dimensions.