Invertible Matrices
What is Invertible Matrix
An array of numbers arranged in the form of rows and columns is known as the matrix. Dimensions of a matrix are given by the number of rows and columns of a matrix, given as m x n where m and n represent the number of rows and columns respectively. The basic mathematical operations like addition, subtraction, multiplication, and division can be done on matrices. There are different types of matrices and have different applications. Here, we will study what is invertible matrix or the invertible matrices.
An invertible matrix is also known as a nonsingular or nondegenerate matrix. An invertible matrix cannot have its determinant value as 0.
Definition of Invertible Matrix
A matrix ‘A’ of dimension n x n is called invertible only under the condition, 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. In such a case matrix B is known as the inverse of matrix A. Inverse of matrix A is symbolically represented by ‘A^{1}‘.
For example, if matrix A and B satisfy this condition AB=BA=I, then we can say B is the inverse of A written as A^{1}=B. Similarly, we can also say A is the inverse of B written as B^{1}
How to Determine if a Matrix is Invertible
An n x n square matrix M is not invertible precisely if det M is 0 which is the determinant value of M is 0, which occurs precisely if the rows (or columns) are not linearly independent, which in turn occurs precisely if the rank of M is not n.
A matrix that has no inverse is singular. When the determinant value of square matrix I exactly zero the matrix is singular.
Invertible Matrix Theorem
Theorem1: Unique inverse is possessed by every invertible matrix.
Proof: Let there be a matrix A of order n×n which is invertible. Let two inverses of A be B and C
Then,AB=BA=In..(1) (In=identity matrix of order n)
and AC=CA=In….(2)
Now,AB=In
=> C(AB)=CIn (premultiplying by C)
=> (CA)B=CIn (by associativity)
=> InB=CIn (since CA=In from (2))
=> B=C. (since InB=B,CIn=C)
So, this proves that an invertible matrix possesses a unique inverse.
Theorem2: A square matrix is invertible if and only if it is nonsingular.
Proof : Suppose there Is an invertble matrix A.Then there exists a matrix B such that
AB=In=BA=>A=In
=> A B=1 ( since AB=AB)
=> A is not equal to 0 => A is nonsingular matrix.
Conversely this can also be proved.
Theorem3: If A and B are invertible matrices and of the same order and are, then (AB)^{1} = B^{1}A^{1}.
Proof:
(AB)(AB)^{1} = I (Taken from the inverse matrix definition)
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)
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}
Invertible Matrix Example
1)Prove that B is inverse of matrix A
A = \[\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}\] B = \[\begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}\]
Solution=> On multiplying A with B we get an identity matrix
AB = \[\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}\] \[\begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}\] = \[\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\]
Similarily when we multiply B with we get an identity matrix
BA = \[\begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}\] \[\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}\] = \[\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\]
Hence we can conclude from above that AB=BA=I
so we can say that B is inverse of A i.e. A^{1}=B
Also, we can say that A is inverse of B i.e.B^{1}=A
2) If A is a nonsingular matrix, then prove that
A^{1}=A^{1} i.e. A^{1}= 1
A
Solution=> Since A is not equal to zero,therefore A^1 exists such that
AA^{1}=I =A^{1}A => AA^{1}=I
=> A A^{1}=1 (sinceAB=AB and I=1)
=> A^{1}= 1
A (since A is not equal to zero)
Matrix Inversion Methods
These are the methods by which we find other matrix B which is the inverse of matrix A and satisfies invertible matrix equation AB=BA=I
The methods are

Gaussian Elimination

Newton’s Method

Cayley Hamilton Method

Eigen Decomposition Method
Applications of Invertible Matrices
The practical applications, where the solution for the system of the equation should be unique there it is necessary that the matrix involved should be invertible. Such applications are:
a) Leastsquares or Regression
b) Simulations
c) MIMO Wireless Communications
1. Is Every NxN Matrix Invertible?
For a matrix to be invertible it must follow the invertible equation that is AB=BA=I. Also, another factor responsible is that the matrix should be nonsingular that is the determinant value of the matrix should not be zero. Every n×n matrix following these conditions is invertible.
This is a continuous function because it is a polynomial in the entries of the matrix. Almost all nbyn matrices are invertible in the language of measure theory. Furthermore, the nbyn invertible matrices are a dense open set in the topological space of all nbyn matrices.
2. What are the Types of Matrices?
Types of matrix differ according to their properties and have different characteristics. Various types of matrices are :
1. Unitary Matrix square matrix whose inverse is equal to its conjugate transpose. It satisfies the condition UH=U^{−1}UH=U^{−1}.
2. Identity Matrix – Identity matrix is a constant matrix having 1 and 0 as its entries. It is a square matrix.
3. Hankel Matrix or catalecticant matrix
4. Hilbert Matrix – square matrix whose entries are the unit fractions. The Hilbert matrix is a special case of the Hankel Matrix.
5. Jacobian Matrix – The Jacobian matrix of a vector function f
6. Transformation Matrix – Transformation matrices have many applications in fields such as computer graphics and the physical sciences. For example, The Transformation matrix needed to rotate an angle θ about the axis defined by the unit vector (l,m,n)
7. Pauli Matrix