Matrix Determinant Calculator
Calculate the determinant of a square matrix.
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How Matrix Determinant Works
The determinant of a square matrix is a scalar value that can be computed from its elements.
For a 2×2 matrix:
\[ \det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc \]
For a 3×3 matrix:
\[ \det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg) \]
For larger matrices, we use cofactor expansion:
\[ \det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} \det(A_{ij}) \]
where A_{ij} is the submatrix obtained by removing row i and column j.
Properties:
- det(AB) = det(A)det(B)
- det(A^T) = det(A)
- det(cA) = c^n det(A) for n×n matrix
- det(A) = 0 if and only if A is singular
- det(A^-1) = 1/det(A) if A is invertible
Applications:
- Solving systems of linear equations
- Finding eigenvalues
- Computing volumes and areas
- Testing for linear independence
- Checking matrix invertibility
Methods for Computing Determinants:
- Cofactor Expansion
- Choose any row or column
- Multiply each element by its cofactor
- Sum the products
- Row Reduction
- Convert to upper triangular form
- Multiply diagonal elements
- Account for row operations
- LU Decomposition
- Factor matrix into L and U
- Multiply diagonal elements of U