Inverse Matrix Calculator
Enter the elements of the square matrix (2x2 to 6x6) to calculate its inverse step by step using the adjoint (adjugate) method.
Quick Examples
How to Use the Calculator
This online inverse 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 answer, the mathematical engine generates a step-by-step resolution so you can understand the entire procedure, using the adjoint (adjugate) matrix method.
Configuration and data entry:
- Select the matrix size: use the top dropdown menu to choose the desired dimension (2x2, 3x3, 4x4, 5x5, 6x6).
- Enter the elements: fill the cells with your values. The calculator supports integers, decimals, and exact fractions (for example, 1/3 or -5/2).
Upon clicking calculate, the tool will process the information and structure the results into two main blocks:
1. Quick answer.
In the first highlighted block, you will get the simplified inverse matrix (denoted as A-1). In case the solver detects that your matrix has a determinant equal to zero (|A| = 0), it will inform you that it is a singular matrix and, therefore, not invertible.
2. Step-by-step resolution.
For 2x2, 3x3, and 4x4 matrices (the most requested dimensions at the academic level), the calculator will display the entire algebraic method to reach the solution. The algorithm will guide you through the following sequence:
- Determinant calculation: the first mandatory step to mathematically verify that the matrix has an inverse (that is, verifying that it is nonzero).
- Adjoint matrix: it will show you the resulting matrix after calculating and transposing all the cofactors.
- Formula application: it will show you how to use the mathematical relationship A-1 = (1 / |A|) · adj(A). You will see how each element of the adjoint matrix is multiplied by the scalar (the reciprocal of the determinant) and how they are simplified to build the final matrix.
Solved Exercises
The following are examples of problems solved by the calculator.
Example 1
Calculate the inverse matrix of the following 3x3 matrix:
$$ \displaystyle A = \begin{bmatrix} 1 & 2 & 3 \\[0.6em] 0 & 1 & 4 \\[0.6em] 5 & 6 & 0 \end{bmatrix} $$
Result
The inverse matrix of A is:
Step-by-step resolution
1. Calculate the determinant.
To determine if the matrix has an inverse, we calculate its determinant. If you want to see the step-by-step calculation, check out the determinant calculator. In this case, the determinant is:
Since the determinant is nonzero, the matrix has an inverse.
2. Calculate the adjoint matrix.
The adjoint (adjugate) matrix is defined as the transpose of the cofactor matrix. In this case, it is:
If you want to see this in more detail, check out the adjoint matrix calculator.
3. Apply the inverse matrix formula.
The inverse matrix is equal to the scalar multiplication of the reciprocal of the determinant and the adjoint matrix, that is:
In our case:
To obtain the final matrix, we multiply each element of the matrix by the scalar and simplify if possible:
Example 2
Find the inverse of the given matrix using the adjoint:
$$ \displaystyle A = \begin{bmatrix} -2 & 1 & 0 \\[0.6em] 3 & -1 & 2 \\[0.6em] 4 & 0 & -3 \end{bmatrix} $$
Result
The inverse of matrix A is:
Step-by-step resolution
1. Calculate the determinant.
To verify if the matrix has an inverse, we find its determinant:
Since the determinant is not zero, we confirm that the matrix is invertible.
2. Calculate the adjoint matrix.
3. Apply the inverse matrix formula.
The inverse matrix is calculated by multiplying the reciprocal of the determinant by the adjoint matrix:
Substituting the values:
We perform the product of the scalar with each entry of the matrix and simplify the results:
Example 3
Determine the inverse matrix of the 2x2 matrix:
$$ \displaystyle A = \begin{bmatrix} 2 & 3 \\[0.6em] 1 & 4 \end{bmatrix} $$
Result
The inverse matrix of A is:
Step-by-step resolution
1. Calculate the determinant.
We evaluate the determinant to check the existence of the inverse:
Since it is different from zero, we conclude that the matrix has an inverse.
2. Calculate the adjoint matrix.
3. Apply the inverse matrix formula.
We obtain the inverse by multiplying the adjoint matrix by the reciprocal of the determinant:
Replacing the data:
Finally, we multiply the constant by each internal element and reduce whenever possible:
Example 4
Compute the inverse matrix of the 4x4 matrix:
$$ \displaystyle A = \begin{bmatrix} 1 & 0 & 2 & -1 \\[0.6em] 3 & 1 & 0 & 2 \\[0.6em] -1 & 2 & 1 & 0 \\[0.6em] 0 & 1 & -2 & 1 \end{bmatrix} $$
Result
The inverse of matrix A is:
Step-by-step resolution
1. Calculate the determinant.
We check if it is possible to invert the matrix by calculating its determinant:
Yielding a value different from zero guarantees the existence of its inverse matrix.
2. Calculate the adjoint matrix.
3. Apply the inverse matrix formula.
We relate the determinant and the adjoint matrix using the corresponding formula:
For this example:
We multiply the scalar fraction by each matrix component and simplify the fractions:
Example 5
Find the inverse (if it exists) of the following matrix:
$$ \displaystyle A = \begin{bmatrix} 2 & 4 \\[0.6em] 1 & 2 \end{bmatrix} $$
Result
Matrix A does not have an inverse because its determinant is zero.
Step-by-step resolution
1. Calculate the determinant.
First, we calculate the determinant of the matrix to verify if an inverse exists:
Since the result is zero, the matrix is singular and therefore does not have an inverse.
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