Matrizen rechnen: Produkt, Inverse, Transponierte
About this app
Matrizenrechnung Schritt für Schritt animiert: A·B als Zeile mal Spalte, wahlweise im Falk-Schema, mit Tausch A·B ↔ B·A; die Inverse über (A | E) mit Gauß-Jordan und exakten Brüchen; die Transponierte als Spiegelung an der Hauptdiagonale.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The idea
In a matrix product every entry comes from one row of A and one column of B: multiply the entries pairwise and add. That is why A needs as many columns as B has rows, and why in general A·B ≠ B·A. The inverse A⁻¹ of a square matrix satisfies A·A⁻¹ = I; you find it by turning (A | I) into (I | A⁻¹) with row operations. The transpose Aᵀ is the reflection in the main diagonal: row i becomes column i.
What you see
For each entry of the product the matching row and column light up, the products are formed one by one and combined into a sum, which then moves to its place in the result. In Falk's scheme B stands on top, A on the left, and row and column cross exactly where the entry belongs. For the inverse the app shows every row operation on (A | I) with exact fractions until the identity stands on the left. When transposing, the entries travel across the diagonal while the diagonal itself stays put.
Things to try
Pick a size or an example, switch Falk's scheme on and off and step forward or back. Clicking an entry of the result jumps to its calculation. Swap A and B and compare the results, or choose sizes that do not fit together. For the inverse, try a matrix that is not invertible and see where the procedure stops. For the transpose, compare a square matrix with a 2×3 matrix.
Frequently asked questions
- How do you multiply two matrices?
- The entry in row i and column j of A·B is the sum of the products of row i of A and column j of B: c_ij = a_i1·b_1j + a_i2·b_2j + … The product is defined only if A has as many columns as B has rows. The result has as many rows as A and as many columns as B.
- What is Falk's scheme?
- A layout for computing by hand: B stands top right, A bottom left, and the result C is built in the field between them. Each entry of C sits exactly where the row of A and the column of B it comes from cross, which keeps you from slipping a row while calculating.
- How do you compute the inverse of a matrix?
- Write A next to the identity as (A | I) and apply row operations until I stands on the left; the right block is then A⁻¹. If a zero row appears on the left, det A = 0 and A is not invertible. The inverse is not obtained by taking 1 over each entry.
- What does the Matrix arithmetic app show?
- It animates three procedures step by step: the matrix product with and without Falk's scheme including the swap A·B ↔ B·A, the inverse via (A | I) with exact fractions and a check, and the transpose as a reflection in the main diagonal.
Subject: Linear algebra
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