On this post we explain what the adjoint of a matrix is and how to find it. Also, you will see several examples of adjoint of matrices and, finally, all the properties of this type of matrix.

Table of Contents

## What is the adjoint of a matrix?

The definition of adjoint of a matrix is as follows:

The **adjoint of a matrix**, also known as **adjugate matrix**, is the transpose of its cofactor matrix.

The adjoint of a matrix is also called **classical adjoint of a matrix** or **adjunct matrix**. Moreover, the adjoint of a matrix is denoted by adj(A).

## How to find the adjoint of a matrix

Once we’ve seen the meaning of the adjoint of a matrix, let’s see how to calculate it:

To **find the adjoint of a matrix**, first replace each element in the matrix by its cofactor and then transpose the matrix.

Remember that the formula to compute the i, j cofactor of a matrix is as follows:

Where M_{ij} is the i, j minor of the matrix, that is, the determinant that results from deleting the i-th row and the j-th column of the matrix.

Note that the adjoint of a matrix can only be found for square matrices.

## Examples of adjoint of matrices

Having seen the theory of the adjoint of a matrix, here are some solved examples of the calculation of the adjoint of a matrix. We will first see the adjoint of a 2×2 dimension matrix, and then the adjoint of a 3×3 dimension matrix.

### Example of a 2×2 matrix

Let A be the following square matrix of order 2:

To compute the adjoint of matrix A, we first have to find the cofactor of each entry of the matrix. To do this, we have to apply the following formula:

Now we replace each element of matrix A by its cofactor to find the cofactor matrix of A:

And finally, we simply have to transpose the cofactor matrix:

On the other hand, there is a formula to find the adjoint of a 2×2 matrix without doing any calculations:

However, this formula is only valid for 2×2 matrices. You can verify the formula by calculating with it the example seen above.

### Example of a 3×3 matrix

Let B be the following square matrix of order 3:

To compute the adjoint of the 3×3 matrix we have to apply the same procedure. So, first we find the cofactor of each element of the matrix:

Secondly, we replace each element of matrix B by its cofactor to determine the cofactor matrix of B:

And finally, we transpose the cofactor matrix to find the adjoint of matrix B:

There is no formula to directly find the adjoint of a 3×3 matrix.

➤ Did you know that the adjugate matrix is used to calculate the inverse of a matrix? See how to find the matrix inverse.

## Properties of the adjoint of a matrix

The adjoint of a matrix has the following characteristics:

- The adjoint of the zero matrix (or null matrix) results in the zero matrix:

- Likewise, the adjoint of the identity matrix of any order results in the identity matrix (of the same order).

- Transposing a matrix first and then finding its adjoint is the same as first finding the adjoint of the matrix and then transposing the result.

- The determinant of the adjoint of a matrix equals to the determinant of the matrix raised to
*n-1*, where*n*is the order of the matrix.

- If matrix A is invertible, then the adjoint of matrix A is equal to the product of the determinant of matrix A and the inverse of matrix A.

- If a matrix is invertible, then we can find the inverse of the adjoint of the matrix using the following formula:

- In addition, if a matrix is invertible, calculating the inverse of the matrix first and then its adjoint is the same as calculating its adjoint first and then inverting the matrix.

- The adjoint of a matrix multiplication equals to the product of the adjoint of each matrix but multiplied in different order:

- The adjoint of a scalar multiplication is equal to the product of the scalar raised to
*n-1*and the adjoint of the matrix, where*n*is the order of the matrix.

- Let A be a square matrix of order
*n*, if the rank of matrix A is less than or equal to*n-2*, then the adjoint of matrix A results in 0.

- Let A be a square matrix of order
*n*, if the rank of matrix A is*n-1*, then the rank of the adjoint of matrix A is 1.