Adjoint Matrix Calculator

Enter the elements of the square matrix (2x2 to 6x6) to calculate its adjoint (adjugate) matrix step by step.

Quick Examples

Rate this tool

How to Use the Calculator

This online adjoint matrix solver is a linear algebra tool designed to process square matrices ranging from 2x2 to 6x6 dimensions. In addition to providing you with the structured final result, the mathematical engine generates a step-by-step resolution to help you understand the logic behind the procedure.

How to enter your data:

  1. Select the matrix size: use the initial configuration menu to define the dimension of the square matrix you are going to work with, choosing any of these formats: 2x2, 3x3, 4x4, 5x5, and 6x6.
  2. Enter the elements: fill in each of the grid cells with your data. The system can process integers, decimals, and exact fractions. To avoid rounding issues during the analytical calculation, we recommend entering fractional values exactly as they appear in your problem.

Once you run the calculation, the tool will organize the results into two main sections:

1. Quick answer
In the first highlighted block, the calculator will directly provide you with the resulting adjoint (adjugate) matrix (denoted as adj(A)).

2. Step-by-step resolution
Following the answer, a summary of the analytical procedure will be displayed. It is important to remember, by mathematical definition, that the adjoint matrix is equal to the transpose of the cofactor matrix. Therefore, the algorithm will structure the general development into two main phases: calculating the cofactor matrix and transposing it.

Solved Examples

The following are examples of problems solved by the calculator.

Example 1

Calculate the adjoint matrix of the 2x2 matrix:

$$ \displaystyle A = \begin{bmatrix} 5 & -2 \\[0.5em] 3 & 4 \end{bmatrix} $$

Result

The adjoint matrix of A is:

$$ \displaystyle \text{adj}(A) = \begin{bmatrix} 4 & 2 \\[0.5em] -3 & 5 \end{bmatrix} $$

Step-by-step resolution

1. Calculate the cofactor matrix.

First, we find the cofactor matrix of A. Its order is the same as the original, and each element is replaced by its corresponding cofactor.

$$ \displaystyle C(A) = \begin{bmatrix} C_{11} & C_{12} \\[0.5em] C_{21} & C_{22} \end{bmatrix} $$

Each cofactor Cij is calculated using the following formula, where Mij is the minor: the determinant of the matrix that results from removing row i and column j from matrix A.

$$ \displaystyle C_{ij} = (-1)^{i+j} M_{ij} $$

Applying the formula to each element of the matrix, we obtain that the cofactor matrix is:

$$ \displaystyle C(A) = \begin{bmatrix} 4 & -3 \\[0.5em] 2 & 5 \end{bmatrix} $$

If you want to see the step-by-step calculation for each cofactor, check out the cofactor matrix calculator.

2. Transpose the cofactor matrix.

The adjoint (adjugate) matrix of A is defined as the transpose of its cofactor matrix. That is, we swap the rows for columns: adj(A) = CT.

$$ \displaystyle \text{adj}(A) = C^T = \begin{bmatrix} 4 & 2 \\[0.5em] -3 & 5 \end{bmatrix} $$
Example 2

Find the adjugate matrix of the 3x3 matrix:

$$ \displaystyle A = \begin{bmatrix} 2 & -3 & 1 \\[0.5em] 4 & 0 & -2 \\[0.5em] -1 & 5 & 3 \end{bmatrix} $$

Result

The adjugate matrix obtained for A is:

$$ \displaystyle \text{adj}(A) = \begin{bmatrix} 10 & 14 & 6 \\[0.5em] -10 & 7 & 8 \\[0.5em] 20 & -7 & 12 \end{bmatrix} $$

Step-by-step resolution

1. Calculate the cofactor matrix.

$$ \displaystyle C(A) = \begin{bmatrix} 10 & -10 & 20 \\[0.5em] 14 & 7 & -7 \\[0.5em] 6 & 8 & 12 \end{bmatrix} $$

2. Transpose the cofactor matrix.

$$ \displaystyle \text{adj}(A) = C^T = \begin{bmatrix} 10 & 14 & 6 \\[0.5em] -10 & 7 & 8 \\[0.5em] 20 & -7 & 12 \end{bmatrix} $$
Example 3

Determine the adjoint matrix of the matrix:

$$ \displaystyle A = \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & -1 \\[0.5em] 3 & -2 & 0 \\[0.5em] \frac{4}{3} & 1 & -\frac{1}{2} \end{bmatrix} $$

Result

The classical adjoint matrix of A is:

$$ \displaystyle \text{adj}(A) = \begin{bmatrix} 1 & -\frac{7}{8} & -2 \\[0.5em] \frac{3}{2} & \frac{13}{12} & -3 \\[0.5em] \frac{17}{3} & -\frac{1}{6} & -\frac{7}{4} \end{bmatrix} $$

Step-by-step resolution

1. Calculate the cofactor matrix.

$$ \displaystyle C(A) = \begin{bmatrix} 1 & \frac{3}{2} & \frac{17}{3} \\[0.5em] -\frac{7}{8} & \frac{13}{12} & -\frac{1}{6} \\[0.5em] -2 & -3 & -\frac{7}{4} \end{bmatrix} $$

2. Transpose the cofactor matrix.

$$ \displaystyle \text{adj}(A) = C^T = \begin{bmatrix} 1 & -\frac{7}{8} & -2 \\[0.5em] \frac{3}{2} & \frac{13}{12} & -3 \\[0.5em] \frac{17}{3} & -\frac{1}{6} & -\frac{7}{4} \end{bmatrix} $$
Example 4

Find the corresponding adjugate matrix for matrix A:

$$ \displaystyle A = \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & -1 \\[0.5em] 3 & -2 & 0 \\[0.5em] \frac{4}{3} & 1 & -\frac{1}{2} \end{bmatrix} $$

Result

The adjugate matrix of A is:

$$ \displaystyle \text{adj}(A) = \begin{bmatrix} 1 & -\frac{7}{8} & -2 \\[0.5em] \frac{3}{2} & \frac{13}{12} & -3 \\[0.5em] \frac{17}{3} & -\frac{1}{6} & -\frac{7}{4} \end{bmatrix} $$

Step-by-step resolution

1. Calculate the cofactor matrix.

$$ \displaystyle C(A) = \begin{bmatrix} 1 & \frac{3}{2} & \frac{17}{3} \\[0.5em] -\frac{7}{8} & \frac{13}{12} & -\frac{1}{6} \\[0.5em] -2 & -3 & -\frac{7}{4} \end{bmatrix} $$

2. Transpose the cofactor matrix.

$$ \displaystyle \text{adj}(A) = C^T = \begin{bmatrix} 1 & -\frac{7}{8} & -2 \\[0.5em] \frac{3}{2} & \frac{13}{12} & -3 \\[0.5em] \frac{17}{3} & -\frac{1}{6} & -\frac{7}{4} \end{bmatrix} $$
Example 5

Obtain the adjoint matrix of the 4x4 matrix:

$$ \displaystyle A = \begin{bmatrix} -1 & -2 & 0 & -3 \\[0.5em] -2 & 1 & -1 & -2 \\[0.5em] -3 & -1 & 2 & 0 \\[0.5em] 0 & -2 & -3 & -1 \end{bmatrix} $$

Result

The resulting classical adjoint matrix is:

$$ \displaystyle \text{adj}(A) = \begin{bmatrix} -15 & 17 & 25 & 11 \\[0.5em] 11 & -25 & 13 & 17 \\[0.5em] -17 & 13 & -3 & 25 \\[0.5em] 29 & 11 & -17 & -15 \end{bmatrix} $$

Step-by-step resolution

1. Calculate the cofactor matrix.

$$ \displaystyle C(A) = \begin{bmatrix} -15 & 11 & -17 & 29 \\[0.5em] 17 & -25 & 13 & 11 \\[0.5em] 25 & 13 & -3 & -17 \\[0.5em] 11 & 17 & 25 & -15 \end{bmatrix} $$

2. Transpose the cofactor matrix.

$$ \displaystyle \text{adj}(A) = C^T = \begin{bmatrix} -15 & 17 & 25 & 11 \\[0.5em] 11 & -25 & 13 & 17 \\[0.5em] -17 & 13 & -3 & 25 \\[0.5em] 29 & 11 & -17 & -15 \end{bmatrix} $$
Example 6

Calculate the adjoint matrix given the matrix:

$$ \displaystyle A = \begin{bmatrix} \frac{1}{2} & 0 & 1 & 2 \\[0.5em] 0 & \frac{3}{4} & 2 & -1 \\[0.5em] 1 & -2 & 0 & \frac{1}{3} \\[0.5em] -1 & 1 & 2 & 0 \end{bmatrix} $$

Result

The adjunct matrix for A is:

$$ \displaystyle \text{adj}(A) = \begin{bmatrix} \frac{25}{6} & \frac{23}{3} & -2 & -\frac{39}{4} \\[0.5em] \frac{8}{3} & \frac{10}{3} & -6 & -\frac{14}{3} \\[0.5em] \frac{3}{4} & \frac{13}{6} & 2 & \frac{19}{8} \\[0.5em] \frac{7}{2} & -3 & -\frac{1}{2} & \frac{5}{4} \end{bmatrix} $$

Step-by-step resolution

1. Calculate the cofactor matrix.

$$ \displaystyle C(A) = \begin{bmatrix} \frac{25}{6} & \frac{8}{3} & \frac{3}{4} & \frac{7}{2} \\[0.5em] \frac{23}{3} & \frac{10}{3} & \frac{13}{6} & -3 \\[0.5em] -2 & -6 & 2 & -\frac{1}{2} \\[0.5em] -\frac{39}{4} & -\frac{14}{3} & \frac{19}{8} & \frac{5}{4} \end{bmatrix} $$

2. Transpose the cofactor matrix.

$$ \displaystyle \text{adj}(A) = C^T = \begin{bmatrix} \frac{25}{6} & \frac{23}{3} & -2 & -\frac{39}{4} \\[0.5em] \frac{8}{3} & \frac{10}{3} & -6 & -\frac{14}{3} \\[0.5em] \frac{3}{4} & \frac{13}{6} & 2 & \frac{19}{8} \\[0.5em] \frac{7}{2} & -3 & -\frac{1}{2} & \frac{5}{4} \end{bmatrix} $$

Related Tools

Daniel Machado

Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.

Last updated: (3 weeks ago)