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.

Adjoint of a Matrix Definition

The adjoint of a square matrix [latex]A=[a_{ij}]_{n\times n}[/latex] is defined as the transpose of the matrix [latex][A_{ij}]_{n\times n}[/latex], where [latex]A_{ij}[/latex] is the cofactor of the element [latex]a_{ij}[/latex]. In other words, the transpose of a cofactor matrix of the square matrix is called the adjoint of the matrix. Adjoint of the matrix A is denoted by [latex]adj\ A[/latex]. 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

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:

[latex]\large A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}[/latex]

Then, the adjoint of this matrix is:

[latex]\large adj\ A=\begin{bmatrix} A_{11} & A_{21}\\ A_{12} & A_{22} \end{bmatrix}[/latex]

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:

[latex]A=\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{22} & a_{33} \end{bmatrix}[/latex]

The adjugate of this matrix is given by:

[latex]\large 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}[/latex]

Here,

[latex]\begin{bmatrix} A_{11} & A_{12} & A_{13}\\ A_{21} & A_{22} & A_{23}\\ A_{31} & A_{22} & A_{33} \end{bmatrix}[/latex] is the cofactor matrix of A.

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.

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:

[latex]\large A = \begin{bmatrix} a_{11} & a_{12} & …. & a_{1n}\\ a_{21} & a_{22} & …. & a_{2n}\\ : & : & …. & :\\ : & : & …. & :\\ a_{n1} & a_{n2} & …. & a_{nn} \end{bmatrix}[/latex]

Now, the adjoint of this matrix is:

[latex]\large 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}[/latex]

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-1

    adj A is invertible with inverse (det A)-1 A

    adj(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 are invertible matrices.

Adjoint of a Matrix Examples

Example 1:

Find the adjoint of the matrix [latex]A =\begin{bmatrix} 2 & 3\\ 1 & 4 \end{bmatrix}[/latex].

Solution:

Given,

[latex]A =\begin{bmatrix} 2 & 3\\ 1 & 4 \end{bmatrix}[/latex]

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, [latex]adj\ A =\begin{bmatrix} 4 & -3\\ -1 & 2 \end{bmatrix}[/latex].

Example 2:

Calculate the adjoint of the matrix [latex]adj\ A =\begin{bmatrix} 1 & -1 & 2\\ 2 & 3 & 5\\ 1 & 0 & 3 \end{bmatrix}[/latex].

Solution:

Given,

[latex]adj\ A =\begin{bmatrix} 1 & -1 & 2\\ 2 & 3 & 5\\ 1 & 0 & 3 \end{bmatrix}[/latex]

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

The cofactor matrix of A [latex]=\begin{bmatrix} 9 & -1 & -3\\ 3 & 1 & -1\\ -11 & -1 & 5 \end{bmatrix}[/latex]

Therefore,

[latex]adj\ A=\begin{bmatrix} 9 & 3 & -11\\ -1 & 1 & -1\\ -3 & -1 & 5 \end{bmatrix}[/latex]

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 [latex]X=\begin{bmatrix} 1 & 2 & 3\\ 0 & 2 & 4\\ 0 & 0 & 5 \end{bmatrix}[/latex].
  2. What is the adjugate matrix of [latex]A=\begin{bmatrix} 1 & 1 & 1\\ 1 & -2 & 3\\ 3 & -1 & 4 \end{bmatrix}[/latex]?
  3. Calculate the adjoint of [latex]A=\begin{bmatrix} -1 & 5\\ 0 & 3 \end{bmatrix}[/latex].

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