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

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.

Adjoint Of a Matrix 2 x 2

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

Then, the adjoint of this matrix is:

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:

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:

The adjugate of this matrix is given by:

Here,

The above formula can be expanded as:

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

Cofactor of the element a_ij = C_ij = (−1)^i+j det(M_ij)

Where, det(M_ij) is called the minor of a_ij.

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, 

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 M_ij be the (n – 1) x (n – 1) matrix obtained by deleting the ith row and jth column. Then, det(M_ij) is called the minor of a_ij. The cofactor C_ij of an element a_ij can be found using the formula:

C_ij = (−1)^i+j det(M_ij)

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:

Now, the adjoint of this matrix is:

Here, A₁₁, A₁₂,…, A₂₁, A₂₂,…., A_nn are the cofactors of the elements a₁₁, a₁₂,…, a₂₁, a₂₂,….,a_nn 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) = k^n-1 adj(A)
• adj(A^T) = (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(A^p) = (adj A)^p
• If A is invertible, then the above formula also holds for negative k.

Click here to understand what invertible matrices are.

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Adjoint of a Matrix Examples

Example 1:

Solution:

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

Example 2:

Solution:

Let C_ij be the cofactor of the element a_ij in matrix A.

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

Cofactors of second-row elements are:

Cofactors of third-row elements are:

Therefore,

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

Adjoint of a Matrix Questions

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