## Introduction

The adjoint of a matrix is the transpose of the cofactor of the matrix. The Adjoint of a Matrix can be used to calculate the inverse of a matrix easily.

In this article, we will discuss the adjoint and inverse of a matrix in detail. So without any further ado, let's start!

## Adjoint of a Matrix

The adjoint of a matrix is the matrix obtained by taking the transpose of the cofactor matrix of a given square matrix. Adj represents the adjoint of any square matrix.

A cofactor is a number obtained by ignoring a specific element's row and column in the form of a square or rectangle.

The adjoint of a matrix can be calculated using the cofactor and transpose of a matrix. It is denoted by adj A.

** adj(A) = (Cofactor of Matrix A) ^{T}**

**Note:** We can find the adjoint of the square matrix only.

Let us understand the adjoint of a matrix with an example:

**Q) Find the Adjoint of a matrix A given below:**

First, we have to calculate all the cofactors of elements in matrix A.

Cofactor of A11 i.e. 4 =

= 1 X 4 - 6 X 2

= -8

Cofactor of A12 i.e. -2 =

= 3 X 4 - 5 X 2

= 2

Cofactor of A13 i.e. 1 =

= 3 X 6 - 5 X 1

= 13

Cofactor of A21 i.e. 3 =

= -2 X 4 - 6 X 1

= -14

Cofactor of A22 i.e. 1 =

= 4 X 4 - 5 X 1

= 11

Cofactor of A23 i.e. 2 =

= 4 X 6 - (-2) X 5

= 34

Cofactor of A31 i.e. 5 =

= -2 X 2 - 1 X 1

= -5

Cofactor of A32 i.e. 6 =

= 4 X 2 - 3 X 1

= 5

Cofactor of A33 i.e. 4 =

= 4 X 1 - 3 X (-2)

= 10

Now, we have to multiply the cofactor of elements with + or - using the formula (-1)^{i+j }. If i+j is even, then we have to multiply it with positive (+) else multiply it with negative (-) sign.

The sign convention for a 3 X 3 matrix is given below:

Now, we will multiply the cofactors with the given sign convention:

After multiplying with the sign, the matrix obtained is:

Now transpose of matrix A is given below:

As we know, the transpose of a matrix's cofactor is its adjoint, so the above matrix is the required answer.