Identity, Inverse, and the Matrix 'Undo' Button

The Matrix Version of One

Just as $a \cdot 1 = a$ for scalars, the identity matrix $I$ has ones on its main diagonal and zeros elsewhere, giving $A I = I A = A$ whenever the shapes match. It represents a transformation that changes nothing.

Transformations That Can Be Undone

For a square matrix $A$ , an inverse satisfies $A A^{- 1} = A^{- 1} A = I$ . A 30-degree rotation is undone by a -30-degree rotation, and uniform doubling is undone by halving. A two-sided inverse exists exactly when the square matrix has full rank, equivalently $det (A) \neq = 0$ in exact mathematics. If a direction is collapsed, distinct inputs share an output and cannot be uniquely recovered.

Rectangular Matrices and the Pseudoinverse

A rectangular matrix has no two-sided inverse of the square-matrix kind. It does not always discard information, however: a full-column-rank map into a higher-dimensional space can preserve distinct inputs and may have a left inverse. A dimension-reducing map generally merges inputs.

The Moore-Penrose pseudoinverse $A^{+}$ handles these cases in a common framework. For an overdetermined system it can select a least-squares solution; when several solutions exist, it can select the minimum-norm one. It is not a promise to reconstruct information that the original transformation discarded.

In Code, Solve the System Directly

Although $X = A^{- 1} B$ is a useful conceptual identity, use np.linalg.solve(A, B) for a square system instead of forming the inverse first. It is generally faster and more numerically stable. For rectangular problems, use a method such as lstsq that matches the question being asked.

An inverse means that a transformation preserved enough information to be undone. In numerical code, solve the system rather than constructing the inverse first.

Code

Concept vs implementation·python

import numpy as np

A = np.array([[2, 1],
              [1, 3]], dtype=float)

# Identity check
I = np.eye(2)
print(np.allclose(A @ I, A))      # True

# Inverse — the concept
A_inv = np.linalg.inv(A)
print(np.allclose(A @ A_inv, I))  # True (modulo float)

# Solving — the implementation
B = np.array([5, 6])
X_via_inverse = A_inv @ B         # textbook, slower
X_via_solve   = np.linalg.solve(A, B)  # what you should actually use
print(np.allclose(X_via_inverse, X_via_solve))   # True

Exercise

Construct a 2×2 matrix that doubles its input. Find its inverse. Verify that the inverse halves vectors. Then construct a singular 2×2 matrix (e.g. one with two identical rows) and try np.linalg.inv — what error do you get?

Hint

Doubling matrix = 2*np.eye(2). Inverse = 0.5*np.eye(2). Singular = [[1, 1], [2, 2]]. The inversion attempt raises LinAlgError: Singular matrix.

The Matrix Version of One

Transformations That Can Be Undone

Rectangular Matrices and the Pseudoinverse

In Code, Solve the System Directly

Code

External links

Exercise

Progress

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