Adjoint Of a Matrix

Before understanding what is an adjoint of a matrix, let’s recall what is a matrix and related terms. In linear algebra, a matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. A matrix having m rows and n columns is called a matrix of order m × n or m × n matrix. However, matrices can be classified based on the number of rows and columns in which elements are arranged. In this article, you will learn about the adjoint of a matrix, finding the adjoint of different matrices, and formulas and examples.

Table of Contents:

Adjoint of a Matrix Definition

The adjoint of a square matrix A = [aij]n×n is defined as the transpose of the matrix [Aij]n×n , where Aij is the cofactor of the element aij. In other words, the transpose of a cofactor matrix of the square matrix is called the adjoint of the matrix. The adjoint of the matrix A is denoted by adj A. This is also known as adjugate matrix or adjunct matrix.

It is necessary to find the adjoint of a given matrix to calculate the inverse matrix. This can be done only for square matrices.

Click here to understand what a square matrix is.

Adjoint of a Matrix Formula

The formula for the adjoint of a matrix can be derived using the cofactor and transpose of a matrix. However, it is easy to find the adjugate matrix for a 2 x 2 matrix. Let’s have a look at the formulas and procedure of finding the adjoint matrix for a given matrix.

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Adjoint Of a Matrix 2 x 2

Let A be the 2 x 2 matrix and is given by:

\(\begin{array}{l} A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}\end{array} \)

Then, the adjoint of this matrix is:

\(\begin{array}{l} adj\ A=\begin{bmatrix} A_{11} & A_{21}\\ A_{12} & A_{22} \end{bmatrix}\end{array} \)

Here,

A11 = Cofactor of a11

A12 = Cofactor of a12

A21 = Cofactor of a21

A22 = Cofactor of a22

Alternatively, the adj A can also be calculated by interchanging a11 and a22 and by changing signs of a12 and a21. This can be shown as:

Adjoint of 2x2 matrix

Learn how to find the cofactor of elements of a matrix here.

Adjoint Of a Matrix 3 x 3

Consider a 3 x 3 matrix as:

\(\begin{array}{l}A=\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{22} & a_{33} \end{bmatrix}\end{array} \)

The adjugate of this matrix is given by:

\(\begin{array}{l} adj\ A=\begin{bmatrix} A_{11} & A_{12} & A_{13}\\ A_{21} & A_{22} & A_{23}\\ A_{31} & A_{22} & A_{33} \end{bmatrix}^T = \begin{bmatrix} A_{11} & A_{21} & A_{31}\\ A_{12} & A_{22} & A_{32}\\ A_{13} & A_{23} & A_{33} \end{bmatrix}\end{array} \)

Here,

\(\begin{array}{l}\begin{bmatrix} A_{11} & A_{12} & A_{13}\\ A_{21} & A_{22} & A_{23}\\ A_{31} & A_{22} & A_{33} \end{bmatrix} \ is\ the\ cofactor\ matrix\ of\ A.\end{array} \)

The above formula can be expanded as:

Adjoint of 3x3 matrix

Alternatively, we can find the cofactors of the matrix using the formula,

Cofactor of the element aij = Cij = (−1)i+j det(Mij)

Where, det(Mij) is called the minor of aij.

What is the Minor?

Minor of an element in a matrix is defined as the determinant obtained by deleting the row and column in which that element lies. e.g. in the determinant of a matrix A, 

\(\begin{array}{l}|A| =\left| \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|\end{array} \)

Minor of a21 is denoted as M21 and is calculated as:

\(\begin{array}{l}{M}_{21}=\left| \begin{matrix} {{a}_{12}} & {{a}_{13}} \\ {{a}_{32}} & {{a}_{33}} \\ \end{matrix} \right|\end{array} \)

What is Cofactor?

A cofactor is a number obtained by eliminating the row and column of a particular element in the form of a square or rectangle. The cofactor is preceded by a negative or positive sign based on the element’s position.

Let A be any matrix of order n x n and Mij be the (n – 1) x (n – 1) matrix obtained by deleting the ith row and jth column. Then, det(Mij) is called the minor of aij. The cofactor Cij of an element aij can be found using the formula:

Cij = (−1)i+j det(Mij)

Thus, the cofactor is always represented with +ve (positive) or -ve (negative) signs.

Also, read: Minors and Cofactors

How To Calculate The Adjoint of a Matrix

We can also write the general formula to find the adjugate of a matrix of order n x n. Let A be the matrix of order n x n, then its adjugate matrix can be written as:

\(\begin{array}{l} A = \begin{bmatrix} a_{11} & a_{12} & …. & a_{1n}\\ a_{21} & a_{22} & …. & a_{2n}\\ : & : & …. & :\\ : & : & …. & :\\ a_{n1} & a_{n2} & …. & a_{nn} \end{bmatrix}\end{array} \)

Now, the adjoint of this matrix is:

\(\begin{array}{l} adj\ A = Transpose\ of\ \begin{bmatrix} A_{11} & A_{12} & …. & A_{1n}\\ A_{21} & A_{22} & …. & A_{2n}\\ : & : & …. & :\\ : & : & …. & :\\ A_{n1} & A_{n2} & …. & A_{nn} \end{bmatrix}=\begin{bmatrix} A_{11} & A_{21} & …. & A_{n1}\\ A_{12} & A_{22} & …. & A_{n2}\\ : & : & …. & :\\ : & : & …. & :\\ A_{1n} & A_{2n} & …. & A_{nn} \end{bmatrix}\end{array} \)

Here, A11, A12,…, A21, A22,…., Ann are the cofactors of the elements a11, a12,…, a21, a22,….,ann respectively.

Adjoint of a Matrix Properties

Some of the important properties of adjugate matrices are listed below.

If A be any given square matrix of order n, we can define the following:

  • A(adj A) = (adj A) A = A I, where I is the identity matrix of order n
  • For a zero matrix 0, adj(0) = 0
  • For an identity matrix I, adj(I) = I
  • For any scalar k, adj(kA) = kn-1 adj(A)
  • adj(AT) = (adj A)T
  • det(adj A), i.e. |adj A| = (det A)n-1
  • If A is an invertible matrix and A-1 be its inverse, then:adj A = (det A)A-1adj A is invertible with inverse (det A)-1 Aadj(A-1) = (adj A)-1
  • Suppose A and B are two matrices of order n, then adj(AB) = (adj B)(adj A)
  • For any non-negative integer p, adj(Ap) = (adj A)p
  • If A is invertible, then the above formula also holds for negative k.

Click here to understand what invertible matrices are.

Watch The Below Video To Understand More About Adjoint of a Matrix

Adjoint of a Matrix Examples

Example 1:

Find the adjoint of the matrix:
\(\begin{array}{l}A =\begin{bmatrix} 2 & 3\\ 1 & 4 \end{bmatrix}\end{array} \)

Solution:

\(\begin{array}{l}Given:\ A =\begin{bmatrix} 2 & 3\\ 1 & 4 \end{bmatrix}\end{array} \)

Here, a11 = 2, a12 = 3, a21 = 1 and a22 = 4.

So the cofactors are:

A11 = a22 = 4

A12 = -a12 = -3

A21 = -a21 = -1

A22 = a11 = 2

Therefore,
\(\begin{array}{l}adj\ A =\begin{bmatrix} 4 & -3\\ -1 & 2 \end{bmatrix}\end{array} \)

Example 2:

Calculate the adjoint of the matrix:
\(\begin{array}{l} A =\begin{bmatrix} 1 & -1 & 2\\ 2 & 3 & 5\\ 1 & 0 & 3 \end{bmatrix}\end{array} \)

Solution:

\(\begin{array}{l}Given:\  A =\begin{bmatrix} 1 & -1 & 2\\ 2 & 3 & 5\\ 1 & 0 & 3 \end{bmatrix}\end{array} \)

Let Cij be the cofactor of the element aij in matrix A.

Now, the cofactors of the elements of the first row are:

Adjoint of a matrix example 2.1

Cofactors of second-row elements are:

Adjoint of a matrix example 2.2

Cofactors of third-row elements are:

Adjoint of a matrix example 2.3

 
\(\begin{array}{l}The\ cofactor\ matrix\ of\ A=\begin{bmatrix} 9 & -1 & -3\\ 3 & 1 & -1\\ -11 & -1 & 5 \end{bmatrix}\end{array} \)

Therefore,

\(\begin{array}{l}adj\ A=\begin{bmatrix} 9 & 3 & -11\\ -1 & 1 & -1\\ -3 & -1 & 5 \end{bmatrix}\end{array} \)

The below practice problems based on the adjoint matrix help in better understanding of the concept.

Adjoint of a Matrix Questions

  1. Find the adjoint of the matrix:
    \(\begin{array}{l}X=\begin{bmatrix} 1 & 2 & 3\\ 0 & 2 & 4\\ 0 & 0 & 5 \end{bmatrix}\end{array} \)
  2. Find the adjugate matrix of A:
    \(\begin{array}{l}A=\begin{bmatrix} 1 & 1 & 1\\ 1 & -2 & 3\\ 3 & -1 & 4 \end{bmatrix}\end{array} \)
  3. Calculate the adjoint of A:
    \(\begin{array}{l}A=\begin{bmatrix} -1 & 5\\ 0 & 3 \end{bmatrix}\end{array} \)

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  1. Great job good examples