Input Matrix (A)
Live Results
Adjoint Matrix: adj(A)
Cofactor Matrix:
How to Use the Calculator
Get your adjoint matrix in three simple steps.
Set Matrix Size
Use the interactive slider to set the dimensions of your square matrix, from 2x2 up to 5x5. The grid will update in real-time.
Input Your Matrix
Fill in the values for each element in the input matrix (A). The calculator will begin computing results instantly as you type.
Analyze the Results
View the determinant and the final adjoint matrix. Toggle "Scientific" mode to also inspect the intermediate cofactor matrix.
What is the Adjoint Matrix?
A key component for finding the inverse of a matrix.
The adjoint (or adjugate) of a square matrix A, denoted as adj(A), is the transpose of its cofactor matrix. The process involves two main steps which our calculator handles for you:
- Find the Cofactor Matrix (C): For each element in the original matrix, we calculate its "cofactor." This is the signed determinant of the smaller matrix (the minor) that remains after removing the element's row and column.
- Transpose the Cofactor Matrix: The rows of the cofactor matrix become the columns of the adjoint matrix, and vice-versa.
The primary use of the adjoint is in finding the inverse of a matrix, according to the formula:
This makes our adjoint matrix calculator a crucial tool for solving systems of linear equations and other advanced applications.
A Professional-Grade Workspace
Features designed for a fast, intuitive, and insightful workflow.
Live Determinant Calculation
Instantly see the determinant of your input matrix update as you type. This key metric is crucial for understanding if a matrix is invertible.
Scientific Mode
For deeper analysis, toggle the Scientific View to reveal the cofactor matrix. This allows you to inspect the intermediate step of the calculation.
Neumorphic Design
The premium, tactile interface with smooth sliders and responsive elements provides a satisfying and clear user experience for complex calculations.
Applications of the Adjoint
Beyond the inverse, the adjoint has several key uses.
Matrix Inversion
The most common application. It provides a direct method to calculate the inverse of a matrix, which is fundamental to solving linear systems Ax = b.
Cramer's Rule
The adjoint matrix is implicitly used in Cramer's Rule, a method for solving systems of linear equations using determinants.
Geometric Transformations
In computer graphics, the transpose of the adjoint matrix (the cofactor matrix) is used to correctly transform surface normals (vectors perpendicular to a surface).
