Adjoint Matrix Calculator

Try out the user-friendly adjoint matrix calculator for calculating the matrix adjoint, and adjugate of a given square matrix.

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Introduction to Adjoint Matrix Calculator:

The adjoint matrix calculator is a great tool for evaluating the adjoint, or adjugate of a given matrix. It is used to find the adjoint of a square matrix A using the cofactor matrix transpose method.

Adjoint Matrix Calculator with steps

Our matrix adjoint calculator is a helpful tool for students, teachers, or professionals who can evaluate adjoint matrix solutions with just one click and a few seconds.

What is Adjoint of a Matrix?

Adjoint of a matrix is a process in which you take the transpose of cofactor matrix elements in linear algebra. The adjoint can be represented with A^T; here, t denotes the transpose of a matrix. It is also known as an adjoint or adjugate matrix.

Although it is a very easy method to find an adjoint matrix, if you know about the matrix basis, especially the cofactor or determinant, you can find the adjoint without any difficulty in manual calculation.

How to Calculate Adjoint of a Matrix?

To find the adjoint of a matrix, you should know about the cofactor determinant method and the transpose of a matrix basis.

Nevertheless, you do not need to worry; we will explain the whole process of finding the adjoint of a matrix with the help of an example of a stepwise process.

Suppose a matrix A. Find the adjoint of the matrix.

$$ A \;=\; \left[ \begin{matrix} 6 & 0 & 1 \\ 3 & 8 & 9 \\ 1 & 4 & 2 \\ \end{matrix} \right] $$

Solution:

$$ Adj(A) \;=\; Adj \left[ \begin{matrix} 6 & 0 & 1 \\ 3 & 8 & 9 \\ 1 & 4 & 2 \\ \end{matrix} \right] $$

Find the cofactor of each element using the minor method.

$$ \left[ \begin{matrix} + \biggr| \begin{matrix} 8 & 9 \\ 4 & 2 \\ \end{matrix} \biggr| & - \biggr| \begin{matrix} 3 & 9 \\ 1 & 2 \\ \end{matrix} \biggr| & + \biggr| \begin{matrix} 3 & 8 \\ 1 & 4 \\ \end{matrix} \biggr| \\ - \biggr| \begin{matrix} 0 & 1 \\ 4 & 2 \\ \end{matrix} \biggr| & + \biggr| \begin{matrix} 6 & 1 \\ 1 & 2 \\ \end{matrix} \biggr| & - \biggr| \begin{matrix} 6 & 0 \\ 1 & 4 \\ \end{matrix} \biggr| \\ + \biggr| \begin{matrix} 0 & 1 \\ 8 & 9 \\ \end{matrix} \biggr| & - \biggr| \begin{matrix} 6 & 1 \\ 3 & 9 \\ \end{matrix} \biggr| & + \biggr| \begin{matrix} 6 & 0 \\ 3 & 8 \\ \end{matrix} \biggr| \end{matrix} \right] $$

Solve all the determinants of all the cofactors of a matrix,

$$ =\; \left[ \begin{matrix} + (8 \times 2 - 9 \times 4) & - (3 \times 2 - 9 \times 1) & + (3 \times 4 - 8 \times 1) \\ -(0 \times 2 - 1 \times 4) & +(6 \times 2 - 1 \times 1) & -(6 \times 4 - 0 \times 1) \\ +(0 \times 9 - 1 \times 8) & -(6 \times 9 - 1 \times 3) & +(6 \times 8 - 0 \times 3) \\ \end{matrix} \right] $$

$$ \left[ \begin{matrix} + (16 - 36) & - (6 - 9) & + (12 - 8) \\ -(0 - 4) & +(12 - 1) & -(24 + 0) \\ +(0 - 8) & -(54 - 3) & +(48 - 3) \\ \end{matrix} \right] $$

After finding the determinate, the new matrix becomes,

$$ \left[ \begin{matrix} -20 & 3 & 4 \\ 4 & 11 & -24 \\ -8 & -51 & 48 \\ \end{matrix} \right] $$

Find the transpose of the above matrix,

$$ \left[ \begin{matrix} -20 & 3 & 4 \\ 4 & 11 & -24 \\ -8 & -51 & 48 \\ \end{matrix} \right]^{T} $$

Therefore, the solution of adjoint matrix A is,

$$ \left[ \begin{matrix} -20 & 4 & -8 \\ 3 & 11 & -51 \\ 4 & -24 & 48 \\ \end{matrix} \right] $$

To skip the manual process and calculate the adjoint of a matrix instantly with step-by-step solutions, try our free matrix adjoint calculator.

How to Use Adjoint of a Matrix Calculator?

The adjoint matrix calculator has a simple design that helps everyone to solve the given matrix adjoint question instantly. You need to put your problem in it and follow some important instructions to get results without any hassle. These instructions are:

Choose the size of the matrix that is given in the input box to find the adjoint of the matrix.
Enter the elements of the matrix that you want to get the solution of the adjoint matrix in the input fields.
Review your given input value to get the exact solution of the adjoint matrix question.
The Calculate button evaluates the example of matrix adjoint problems.
If you want to check the workings behind our adjoint of 3x3 matrix calculator, you can use the load example and its solution to learn more about this concept.
Click the “Recalculate” button for the evaluation of more examples of the adjoint matrix with the solution.

What Outcome Adjoint of Matrix Calculator Give?

The adjoint of a matrix calculator provides you with a solution per your input problem when you click the calculate button. It may include the following:

In the Result Box,

Click on the result button to get the solution to your adjoint matrix question.

Steps Box

When you click on the steps option, you get the solution to the adjoint matrix questions step-by-step.

Useful features of Adjoint Calculator:

The adjoint of a matrix calculator has many useful features. It allows you to solve adjoint matrix problems and get their solutions immediately.

Our tool only gives the input value and provides a solution without manual calculation. These features are:

It is a reliable tool as it always provides you with accurate solutions of the given matrix into an adjoint matrix
The adjoint of 3x3 matrix calculator is an efficient tool that provides solutions in the form of adjoint matrices from the given matrix problems in a few seconds.
It is a learning tool that provides you with complete information about the concept of adjoint matrix very easily through online platforms.
Adjoint matrix calculator is a free tool that allows you to use it for the calculation of adjoint matrix without taking a fee.
It is a handy tool that solves adjoint matrix problems quickly without external effort.
Adjoint of matrix calculator is an easy-to-use tool; anyone or even a beginner can easily use it for the solution of matrix adjoint problems.

Related References

How would you define adjoint of matrix?
Explain the evaluation process of adjoint of matrix:
Give some examples of adjoint matrix:
Give an overview of adjoint of a matrix?

Frequently Ask Questions

Are hermitian and adjoint the same matrix?

Yes, Hermitian and Adjoint matrices are the same. The terms "Hermitian" and "adjoint" are used interchangeably, but they are used in different contexts in linear algebra. Here’s the difference are

Hermitian Matrix:

A matrix A is hermitian if it is equal to its own conjugate transpose A = A^h where A^h is the conjugate transpose of A. For a complex matrix A, A^h is obtained by taking the transpose of A and then taking the complex conjugate of each entry.

Adjoint Matrix:

In linear algebra, a matrix A is called adjoint when you take the transpose of cofactor elements matrix A. It is represented as A^t here t is the obtained after transposition of matrix A.

What is the difference between an adjoint and inverse matrix?

The adjoint (or adjugate) matrix and the inverse matrix are related but distinct concepts in linear algebra. Their differences are:

Adjoint (Adjugate) Matrix

Adjoint of a matrix A is a process in which you take the transpose of cofactor matrix only.

Inverse Matrix

Inverse of matrix A is denoted with matrix A^-1 such that, A^-1 A = I, where I is the identity matrix. The inverse exists only if A is square and has full rank and its determinant is non-zero.

Hence, the adjoint matrix exists for any square matrix, and the inverse matrix exists only if the matrix is square and invertible. On the other hand, the adjoint matrix uses the cofactors of the original matrix but the inverse matrix uses various methods, including Gaussian elimination, adjoint method, or LU decomposition.

Does a matrix commute with its adjoint?

No, a matrix does not necessarily commute with its adjoint (adjugate). However, there are special cases where they commute.

Adjoint (Adjugate) Matrix: The adjoint (or adjugate) of a matrix is the transpose of the cofactor matrix of A.

Commute law: Two matrices A and B commute if AB = BA.

Commutativity of a Matrix with its Adjoint for special case:

To determine whether a matrix A commutes with its adjoint adj(A), we need to check if:

Diagonal Matrices:

For diagonal matrices, A and adj(A) do commute. If A is a diagonal matrix, its adjugate is also a diagonal matrix, because the diagonal matrices commute with each other.

Identity Matrix:

For the identity matrix I:

$$ adj(I) \;=\; I $$

How to find inverse of matrix using adjoint method?

To find the inverse of a matrix we use the adjoint (adjugate) method that has a few steps. Here's a detailed explanation of its process.

Calculate the Cofactor Matrix:

For each element aij in the matrix A, compute its cofactor. The cofactor Cij is given by: Cij = (−1)i + j ⋅ det(Mij)

where Mij is the (n−1)×(n−1) submatrix obtained from the i-th row and j-th column from A.

Form the Cofactor Matrix:

Construct the cofactor matrix C by placing each cofactor Cij in the corresponding position (i,j).

Transpose the Cofactor Matrix:

The adjugate (adjoint) matrix adj(A) is the transpose of the cofactor matrix C: adj(A) = C^T

Calculate the Determinant of A:

Compute the determinant det⁡(A) of the original matrix A.

Find the Inverse:

The inverse of A is given by $$ A^{−1} \;=\; \frac{1}{det⁡(A)*adj(A)} $$

This method works for any square matrix with a non-zero determinant.

A symmetric matrix is self adjoint or not?

Yes, a symmetric matrix is self-adjoint because it satisfies the property of a symmetric matrix.

Symmetric Matrix:

A matrix A is symmetric if it equals its transpose: A = A^T

Self-Adjoint Matrix (Hermitian Matrix):

A matrix A is self-adjoint (or Hermitian) if it equals its conjugate transpose: A = A^h

Symmetric Matrices and Self-Adjoint Property

For a real matrix A, the conjugate transpose A^h is just the transpose A^T. Thus, the definition of a symmetric matrix (where A = A^T) coincides with the definition of a self-adjoint matrix.

Amanda Jepson

Last Updated February 21, 2024