Determinant Calculator
What is a Determinant?
A determinant is a scalar value that can be computed from the elements of a square matrix. It helps determine many properties of the matrix, such as whether the matrix is invertible, and is used in solving systems of linear equations, finding eigenvalues, and calculating volumes in multidimensional spaces.
How to Calculate a Determinant
The method for calculating a determinant depends on the size of the matrix:
For a 2×2 Matrix
If we have a matrix A = [[a, b], [c, d]], the determinant is calculated as:
det(A) = ad - bc
For a 3×3 Matrix
For a 3×3 matrix, we can use the method of cofactor expansion or the Sarrus rule.
For matrix A = [[a, b, c], [d, e, f], [g, h, i]],
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
For Larger Matrices
For 4×4 and larger matrices, we typically use cofactor expansion, which breaks down the calculation into a series of smaller determinants. Our calculator handles this complexity for you automatically.
Applications of Determinants
- Solving systems of linear equations using Cramer's rule
- Finding the inverse of a matrix
- Calculating the area of a triangle or parallelogram
- Computing the volume of a parallelepiped
- Determining if vectors are linearly independent
- Finding eigenvalues in linear transformations
Tips for Using the Calculator
- Select the matrix size first (2×2, 3×3, or 4×4)
- Enter all values in the matrix cells
- Click "Calculate" to find the determinant
- Use the "Reset" button to clear all values and start over
- For fractions or decimals, enter them directly (e.g., 1/2 or 0.5)