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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.

 

For example, matrices A and B are given below:

Now we multiply A with B and obtain an identity matrix:

Similarly, on multiplying B with A, we obtain the same identity matrix:

It can be concluded here that AB = BA = I. Hence A^-1 = B, and B is known as the inverse of A. Similarly, A can also be called an inverse of B, or B^-1 = A.

A square matrix that is not invertible is called singular or degenerate. A square matrix is called singular if and only if the value of its determinant is equal to zero. Singular matrices are unique in the sense that if the entries of a square matrix are randomly selected from any finite region on the number line or complex plane, then the probability that the matrix is singular is 0, that means, it will “rarely” be singular.

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.

Theorem 2:

If A and B are matrices of the same order and are invertible, then (AB)^-1 = B^-1 A^-1.

Proof:

(AB)(AB)^-1 = I (From the definition of inverse of a matrix)

A^-1 (AB)(AB)^-1 = A^-1 I (Multiplying A^-1 on both sides)

(A^-1 A) B (AB)^-1 = A^-1 (A^-1 I = A^-1 )

I B (AB)^-1 = A^-1

B (AB)^-1 = A^-1

B^-1 B (AB)^-1 = B^-1 A^-1

I (AB)^-1 = B^-1 A^-1

(AB)^-1 = B^-1 A^-1

Matrix Inversion Methods

• There are a variety of matrix inversion methods, which can be broadly divided into two categories: exact and approximate. Exact methods are those that produce a result that is mathematically correct, while approximate methods produce a result that is not mathematically perfect, but is still close to the correct value.One common exact matrix inversion method is the Gauss-Jordan elimination method. This approach begins by transforming the matrix into an upper triangular form, which makes it easier to invert. The inverse is then found by solving a set of linear equations.

  Another exact method is the LU decomposition, which breaks the matrix into a product of two lower triangular matrices. The inversion is then performed on each of these smaller matrices.

  Approximate methods include the QR decomposition and the Cholesky decomposition. The QR decomposition uses orthogonal matrices to approximate the inverse, while the Cholesky decomposition uses a symmetric matrix to approximate the inverse. Both of these methods are relatively fast and efficient, and are often used when the exact inverse is not required.