Nonsingular Matrix

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

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◇Overview

Definition: An n×n square matrix A is called nonsingular or invertible if there exists an n×n matrix B such that:

◇Key Concepts

Singular Matrix Uniqueness

◇Details / Theory

Definition: An n×n square matrix A is called nonsingular or invertible if there exists an n×n matrix B such that:
AB=BA=In​
The matrix B is called the inverse of A and is denoted by A−1. (Handout 2, Page 12) Singular Matrix: If such a matrix B does not exist, A is called singular or noninvertible. Uniqueness: If a matrix has an inverse, its inverse is unique. (Handout 2, Page 13)

Properties of Inverses

If A and B are nonsingular n×n matrices and k is a non-zero scalar: (Handout 2, Page 14)

1. Inverse of Inverse: (A−1)−1=A

2. Inverse of a Product: (AB)−1=B−1A−1 (Note the reversal of order)
   Personal Note: This is similar to the transpose of a product: (AB)⊤=B⊤A⊤.

3. Inverse of a Scalar Multiple: (kA)−1=k1​A−1

4. Inverse of a Power: (Am)−1=(A−1)m, often written as A−m, for non-negative integer m.

5. Inverse of Transpose: (A⊤)−1=(A−1)⊤ (Handout 2, Page 15)

Finding the Inverse (Brief Mention - methods are more complex)

For a 2×2 matrix A=[a​b c​d​]:
If ad−bc=0 (this is the ◇determinant), then A is invertible and:
A−1=ad−bc1​[d​−b −c​a​]
(Handout 2, Page 12, Example 10 shows this) For larger matrices, methods like ◇Gauss-Jordan Elimination (using ◇Elementary Row Operations on an augmented matrix [A∣I] to get [I∣A−1]) are used.

◇Common Pitfalls

"Nonsingular" = "invertible" — they are the same thing. A matrix with a row/column of zeros is always singular ($\det = 0$).

◇Summary

TL;DR — A nonsingular (invertible) matrix has $\det(A) \neq 0$, is row-equivalent to $I$, and gives a unique solution to $A\mathbf{x} = \mathbf{b}$ for every $\mathbf{b}$.

◇Links

Related: - ◇Determinant of a Matrix - ◇Gauss-Jordan Elimination - ◇Elementary Row Operations Source / Reference:

◇References & Navigation

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• Related Notes: ◇Lower Triangular Matrix, ◇Identity Matrix, ◇Matrices Detailed