Matrices like numbers: identity, inverse, A·x = b
About this app
Who plays the role of 0 and 1 for matrices? The zero matrix for plus, the identity for times, not the matrix full of ones: (2 3; 5 1) times the all-ones matrix gives (5 5; 6 6). Opposite matrix and inverse as reciprocal, A·x = b unfolds into three equations, x = A⁻¹·b, Gauss-Jordan on (A | I) and the phone matrix 1 to 9, which has no inverse (0 = 2).
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The idea
Matrices can be added and multiplied, so just as for numbers one asks for neutral and inverse elements. The zero matrix is neutral for addition. For multiplying square matrices the neutral element is not the matrix full of ones but the identity matrix with ones on the main diagonal: A·I = I·A = A. The inverse A⁻¹ is the reciprocal of a matrix, A⁻¹·A = I. With it a system of linear equations A·x = b can be solved as x = A⁻¹·b, provided the inverse exists.
What you see
Five topics play step by step, with exact fractions. (2 3; 5 1) times the all-ones matrix gives (5 5; 6 6); times the identity A stays. For the opposite matrix and A⁻¹ of (4 7; 2 6) the check is computed entry by entry. The phone matrix 1 to 9 times x unfolds into three equations. The formula x = A⁻¹·b comes from multiplying from the left. Gauss-Jordan turns (A | I) into (I | A⁻¹); for the phone matrix a zero row appears instead.
Things to try
Pick a topic and step through it with Next and Back, or play it all. Every step has a sentence explaining what happens. In Gauss-Jordan you switch between an invertible matrix and the phone matrix, once with a right-hand side b and once with I, and see where the procedure fails.
Frequently asked questions
- What is the identity matrix?
- The square matrix with ones on the main diagonal and zeros elsewhere. It is the neutral element of matrix multiplication: A·I = I·A = A. The matrix in which every entry is 1 is not; it sums up the rows.
- When does a matrix have an inverse?
- Only square matrices can have one, and not all of them do. A is invertible exactly when det A ≠ 0. In the Gauss-Jordan procedure on (A | I) this shows as follows: if a zero row appears on the left, there is no inverse.
- How do you write a system of equations as A·x = b?
- The coefficients go into the matrix A, the unknowns into the vector x, the right-hand sides into the vector b. A·x is an ordinary matrix multiplication, and two vectors are equal if all components are equal. That gives back the individual equations.
- What does the Matrices like numbers app show?
- It animates neutral elements (zero matrix, all-ones matrix, identity), the opposite matrix and the inverse, the translation of A·x = b into three equations, the solution x = A⁻¹·b and the Gauss-Jordan procedure, including a matrix without an inverse.
Subject: Linear algebra
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