Adjoint, Inverse of Square Matrix

In this part of the introduction, before directly studying the application of matrices on different operations like Adjoint of Matrix and Inverse of Square Matrix, we have to know about the Matrix; what is a matrix? The exact rectangular arrangement of numbers in particular rows and columns, enclosed in a [] or (), is called the matrix of order a by b. They are effectively used in various accounting, economics, business, finance, and statistics. Adjoint of Matrix: The transpose of the cofactor matrix of the A is called the adjoint of the square matrix.

Operations On Matrix

The inverse of Square Matrix: Suppose M is the square matrix of order n, there also exists another square matrix N of the same order as of A such that MN=NM=I, where I is the unit matrix of order m then the square matrix N is called the inverse of the square matrix M and it is denoted by M-1 (it is read as the inverse of M or M inverse).

 Using the notation A-1 for B, we can write the above equations as AA-1=A-1A= I.

 Note: For the existence of inverse of matrix A, it is necessary that |A| ≠ 0, which means A is the nonsingular matrix.

Properties of the Matrix:

●      The order in the matrix is denoted as m×n, and it’s read as m by n.

●      Each member in the matrix is called an element of that matrix.

●      A matrix by itself does not have any value or any special meaning.

●      Matrices are usually denoted by capital letters such as A, B, C.

●      And the elements of the matrix are denoted by the small letters aij, bij, cij, etc. are the elements of ith row and jth column of the matrix A.

For example, i) A= [  2 -1   3

                            1   0  -5

                             4 -2   1  ]

here a32 = -2

In the above example, A is the matrix having three rows and three columns, where A’s order is 3×3. And there are nine elements in matrix A.

Square Matrix: 

It is the matrix in which the number of rows equals the number of columns. Suppose the square matrix has order n×n, then the n is called the order of the square matrix.

 Points to be noted :

1. Let A = [aij ]m×m be a square matrix of order m, then the elements of this square matrix are a11, a22,a33,…., aii, …, amm are called as the diagonal elements of a square matrix A. Here’s the point to be remembered: diagonal elements are defined only for the Square Matrix.

2. In elements mij, Where i≠j are called non-diagonal elements.

3. In elements mij, Where i

4. Elements mij, where i>j, are called elements below the diagonal.

We can verify statements 3 and 4 by looking at the matrices of a different order.

 The particularity of the adjoint of the inverse of a matrix -:

 It is possible to prove that if A is a square matrix where |A|≠0, its inverse, i.e., A-1 is unique.

Now here, we are going to prove this uniqueness of the inverse of a square matrix by the theorem below:

Mathematics  Theorem: Prove the uniqueness of the inverse of a square matrix; suppose it’s M.

Proof: Let’s take M as a square matrix having order n, and let its inverse exist.

Let’s take N and the two inverses of the square matrix M.

Then by considering the definition of the inverse matrix,

MN=NM=I and MO=OM=I

Now consider= N=NI

                   =N(MO)

                   =(NM)O

                   =IO

              N = O

Hence it is proved that N = O, i.e., if the inverse of a matrix exists, it’s always unique.

 Adjoint and inverse square matrix problems are based on the above theorem.

There are only two methods of obtaining the inverse( if it exists) of the square matrix as follows:-

1. Elementary Transformations

2. Adjoint Method

Both the above methods can easily obtain the inverse of a square matrix.

 Firstly here we are going to see the Adjoint method below:- 

●      This is the best method of finding the inverse, which can be directly used for obtaining the inverse of any square matrix.

●      For understanding such a method, we have to know the definitions of the minor and cofactor.

●      Both the minor and cofactor are essential parts of the square matrix for obtaining its inverse with the help of the adjoint method.

Adjoint and inverse of square matrix example: 

if one calculates the values for 2A=A*2 (multiply each entry of matrix A by 2 and obtain the results in a new matrix), it should be noted that only the values on the diagonal of A are unchanged. Likewise, the changes to the value of 2A are only at the (1,1) location on the diagonal. In matrix A there are three distinct vectors that can have their entries switched. Therefore, as noted by Nelson and Moser (1999), it comes natural to treat these 3 vectors as a 3-vector A’ (rather than just a pair of 2-vectors). The choice of which 3-vector to treat as adjoint is controlled by the rule of order. If that rule is a permutation, then the matrix A’ must also be a permutation. This is satisfied when A’ = P*Q (where P and Q are themselves column permutations). For example, if A = [1 2 3 4] and the rule given above submits one of the rows to be swapped, then matrix B to be formed must be [0 1 4 3].

Conclusions:

With this, we had mentioned that there are two methods of obtaining the inverse of the square matrix but only focused on one, which is an adjoint method. Another method of the inverse of the square matrix is the Elementary transformation of a square matrix. Elementary transformation is not as direct a method as an adjoint method. It is divided into some steps that should be followed to obtain the inverse of a square matrix.