Matrix Determinant Calculator | 2x2, 3x3, NxN Determinants
Calculate the determinant of a square matrix.
How to Use
- Enter the elements of a square matrix in the text area.
- Ensure the number of rows matches the number of columns (e.g., 2x2, 3x3, 4x4).
- Use spaces or commas to separate numbers in a row, and use new lines for each row.
- Click 'Calculate Determinant' to compute the value.
About Determinants
The determinant is a special scalar value defined only for square matrices that contains significant information about the matrix and the linear transformation it represents.
Geometric Meaning
In geometry, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. If the determinant is zero, the volume is zero, indicating the vectors are linearly dependent.
Invertibility
A square matrix is invertible (non-singular) if and only if its determinant is non-zero. If the determinant is exactly zero, the matrix is singular and does not have an inverse.
Properties of Determinants
- The determinant of the identity matrix is 1: det(I) = 1.
- The determinant of a transpose is the same: det(Aᵀ) = det(A).
- Multiplicativity: det(AB) = det(A)det(B).
- Swapping two rows (or columns) negates the determinant.
Applications
- Solving systems of linear equations (Cramer's Rule).
- Finding the inverse of a matrix.
- Determining linear independence of a set of vectors.
- Calculating volumes and areas in calculus (Jacobian determinant).