Adjoint and Inverse of a Matrix

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What is the Adjoint of a Matrix?

The adjoint of matrix A = [aij]n x n  is mathematically equated as the transpose of the matrix [Aij]n x n, where Aij is the cofactor of the element aij. 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.


Matrices - Operations


Adjoint Matrices

A cofactor matrix C of a matrix A is the square matrix of the same order as A in which each element aij is replaced by its cofactor cij.


Example:


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


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


Define Adjoint of a Matrix

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 when switching the elements of the row with columns and the elements of a column with the row. This is represented by AT.


For e.g.: A= [2 3]

               AT\[\begin{bmatrix}2\\3 \end{bmatrix}\]

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?

(i) Adjoint matrix of 2 x 2

To find 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.


(ii) Adjoint of a matrix 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:

[A][B]=[B][A]=[I]  where I is an identity matrix.


How do we Establish a Relation Between Adjoint and 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|


Properties of Adjoint and Inverse of a Matrix

(i) If A is any given matrix of order nxn, then 

A adj(A) = adj(A) A = |A|I, 

where I denote the identity matrix of order n.


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


(iii) 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.


(iv) A square matrix A can be invertible if and only if A is a non-singular matrix.


Concept of Matrices

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 which 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= aij where the element is in the ith row and jth column. Knowing what exactly is a matrix 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.


E.g: 2 3 is a matrix where there are 1 row and 2 columns. This can also be expressed in the form 1x3 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

FAQ (Frequently Asked Questions)

1) What is a Matrix?

Ans) Matrices, which is a plural form of the matrix, is a way of expressing numbers, symbols, expressions, etc. It is arranged in rows and columns as well as placed in the box brackets. 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.


E.g.: 2 3 is a matrix where there are 1 row and 2 columns. 

This can also be expressed in the form 1x3 matrix. 

Each digit, symbol, expression, etc. is called the element of the matrix.

2) How to Calculate the Adjoint of a Matrix?

Ans) As already mentioned, the adjoint of matrix A = [aij]n x n  is mathematically expressed as the transpose of the matrix [Aij]n x n, where Aij is the cofactor of the element aij. Adjoining of the matrix A is denoted by adj A. We can only find the adjoint of a square matrix. To calculate the adjoint do the following steps -


(i) adjoint matrix of 2x2

To find 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.


(ii) adjoint of a matrix 3X3  

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.