Linear Algebra Visual Tools

Watch the element-wise product A circle-dot B build one cell at a time, with animated arrows linking each pair of operand cells to its destination. Adjust the shared shape of A and B from 1x1 to 5x5 and see how the Hadamard product differs from the standard matrix product.

Watch A + B = C or A - B = C build one cell at a time, with animated arrows linking each pair of operand cells to its destination in the result matrix. Toggle between addition and subtraction, adjust the shared shape of A and B from 1x1 to 5x5, and step through the element-wise rule at your own pace.

Watch A become A^T through four equivalent methods — cell-by-cell, row-as-column, column-as-row, or a single diagonal reflection. Set the shape of A from 1x1 up to 5x5, switch methods with a tabbed control panel, and see how each mental model breaks the same operation into a different number of steps.

Transform any matrix to row echelon form (REF) or reduced row echelon form (RREF) one row operation at a time. Choose a size from 2×3 up to 5×6, fill the cells by hand or generate random values, then pick which form to reduce to. Each stage shows the operation performed — row swap, scaling, or elimination — with the affected rows highlighted in the matrix and the row state before and after spelled out in textbook notation. Step manually with Previous and Next, or hit Play for automatic 1-second advance. RREF makes solutions immediately readable; REF stops earlier and leaves back-substitution to you.

Step-by-step animated matrix multiplication with full control over the calculation. Watch each dot product form element-by-element, with red highlighting on the specific pair of values currently being multiplied and yellow on the full row-and-column being combined. Step forward and back, pause, resume, or let it auto-play. Resize either matrix from 1×1 up to 10×10, generate random values within any range, or edit cells by hand — the result matrix dimensions update automatically, and incompatible dimensions are caught before computation starts. A running commentary below the grids spells out every multiplication and the accumulating sum, so the formula C[i][j] = Σ A[i][k]·B[k][j] becomes concrete arithmetic you can verify by hand.

Watch tr(A) build one diagonal entry at a time, with off-diagonal cells fading into the background. Set the square dimension of A from 2x2 up to 10x10, follow the running sum, and see why the trace is the cleanest scalar invariant of a square matrix.

Pair, multiply, sum — watch the inner product build into a single scalar one term at a time. Switch between the classical vector dot product and the Frobenius matrix inner product, set the shape of the operands, and follow a running expanded sum with per-term color coding.