Adjoint and Inverse of a Matrix

Matrix is a mathematical concept that is used to solve linear equations primarily. It was called Arrays until the 1800s. The term ‘matrix’ was used by James Joseph Sylvester in 1850. Matrix is derived from a Latin word that means ‘womb’. Later in 1913 an English mathematician, Cullis used the box bracket and also developed a way to express matrices. The expression to show an element of the matrix is A= a_ij where the element is in the i^th row and j^th column. Knowing what exactly a matrix is and its uses make it all easy to understand the adjoint and inverse of a matrix.

Matrices are defined by rows and columns. A general way of expressing matrices is m x n or m by n where m is the number of rows and n is the number of columns.

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Define Adjoint of a Matrix

The adjoint of matrix A = [a_ij]_nxn  is mathematically equated as the transpose of the matrix [A_ij]_nxn , where A_ij is the cofactor of the element a_ij. Adjoining of the matrix A is denoted by adj A. We can only find the adjoint of a square matrix. Let’s discuss the adjoint of a matrix example to understand this clearly.

To define the adjoint of a matrix, first, we need to understand another term called transpose of a matrix and cofactors. Transpose of a matrix means switching the elements of the row with columns and the elements of a column with the row. This is represented by A^T.

For e.g. A=23

\[A^{T} = \begin{bmatrix}2 \\3 \end{bmatrix}\]

A cofactor is a number you get when you remove the row and column of a designated element in a matrix, which is just a numerical grid.

How to Find the Adjoint of a Matrix?

There is no particular formula and method of calculation, you can find the adjoint of matrix by following a few basic steps for different types of the matrix: 

1. Adjoint matrix of 2 x 2

To find the adjoint of a matrix, you simply have to swap elements a₁₁ with a₂₂ and switch the signs of elements a₁₂ and a₂₁ from positive to negative or vice versa.

2. Adjoint matrix of 3 X 3  

To find the adjoint of a 3x3 matrix, you will have to find the cofactors of each element. After finding the cofactors, arrange them in a matrix. Transpose of this matrix will simply give you the adjoint of the matrix. 

What is the Inverse of a Matrix?

A matrix B will be called the inverse of matrix A when the product of these matrices gives an identity matrix. An identity matrix is a matrix where all the diagonal elements are 1 and the other elements are 0.

It can be expressed in the following way in mathematical terms:

AB = BA  = I  where I is an identity matrix.

How do we Establish a Relation between the Adjoint and the Inverse of a Matrix?

We can find the inverse of a matrix by simply dividing the adjoint of the matrix by the determinant of the matrix.

A^-1 =  adj A|A|

Matrices - Operations

Adjoint Matrices

A cofactor matrix C of matrix A is the square matrix of the same order as A in which each element a_ij is replaced by its cofactor c_ij.

If A = \[\begin{bmatrix}1 & 2 \\-3 & 4 \end{bmatrix}\]

The cofactor C of A is C = \[\begin{bmatrix}4 & 3 \\-2 & 1 \end{bmatrix}\]

Properties of Adjoint and Inverse of a Matrix

1. If A is any given matrix of order n x n, then 

A adj(A) = adj(A) A = |A|I, 

Here I denote the identity matrix of order n.

2. If B and A are non-singular matrices of the same order, then AB and BA will also be non-singular matrices of the same order.

3. The product of the determinants is equal to the determinant of the product matrices, that is,|A||B| = |AB|, where A and B are square matrices of the same order.

4. A square matrix A can be invertible if and only if A is a non-singular matrix.

e.g.: \[[2 \, 3]\] is a matrix where there are 1 row and 2 columns. This can also be expressed in the form 1 x 2 matrix. Each digit, symbol, expression, etc., is called the element of the matrix. Several operations can be done when both the matrices are of the same dimension. Operations like:

• Addition

• Subtraction

• Multiplication

In this article, we have learned how to find the adjoint of a matrix. Also, we learned the method to find the inverse of a matrix using the adjoint of that matrix.