This document discusses invertible matrices and their properties. An invertible or non-singular matrix is a square matrix that has an inverse matrix such that when multiplied together they form the identity matrix. The inverse of a matrix is unique. Invertible matrices have applications in simulations and wireless communications where unique solutions are needed. Properties of invertible matrices include that the inverse of the inverse is the original matrix and the inverse of a product of matrices is the product of the inverses in reverse order. The document also mentions LU decomposition which factors a matrix into lower and upper triangular matrices.

1. Invertible Matrix and
Factorization

2. What is Invertible Matrix?
 A matrix A of dimension n x n is called invertible
 if and only if there exists another matrix B of the same dimension,
 such that AB = BA = I, where I is the identity matrix of the same order. Matrix
B is known as the inverse of matrix A.
 Inverse of matrix A is symbolically represented by A-1. Invertible matrix is also
known as a non-singular matrix or nondegenerate matrix.

3. Example

5. Example

8. Invertible Matrix Theorem
 Theorem 1
 If there exists an inverse of a square matrix, it is always unique.
 Proof:
 Let us take A to be a square matrix of order n x n. Let us assume matrices B
and C to be inverses of matrix A.
 Now AB = BA = I since B is the inverse of matrix A.
 Similarly, AC = CA = I.
 But, B = BI = B (AC) = (BA) C = IC = C
 This proves B = C, or B and C are the same matrices.

9. Applications of Invertible Matrix
 For many practical applications, the solution for the system of the equation
should be unique and it is necessary that the matrix involved should be
invertible. Such applications are:
 Simulations ( create real time like events like pilot training simulator
 MIMO Wireless Communication (antenna Technology Used in
Telecommunication

10. Properties
 Below are the following properties hold for an invertible matrix A:
 (A−1)−1 = A
 (kA)−1 = k−1A−1 for any nonzero scalar k
 (AT)−1 = (A−1)T
 For any invertible n x n matrices A and B, (AB)−1 = B−1A−1. More specifically,
if A1, A2…, Ak are invertible n x n matrices, then (A1A2⋅⋅⋅Ak-1Ak)−1 =
A−1kA−1k−1⋯A−12A−11
 det A−1 = (det A)−1

11. LU Decomposition
 In linear algebra, LU Decomposition, i.e., lower–upper (LU) decomposition or
factorization of a matrix, can be defined as the product of a lower and an
upper triangular matrices.