Matrix Determinant Calculator
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The determinant (determinant) of a matrix is a scalar number calculated from a square matrix, key in linear algebra: it checks the invertibility of the matrix, calculates volumes and geometric transformations. The calculator calculates the determinant showing steps for 2×2, 3×3 and 4×4.
2×2 matrix determinant
Matrix [[a,b],[c,d]]. det = ad - bc. Example: [[3,1],[2,4]]. det = 3×4 - 1×2 = 12 - 2 = 10. Geometric interpretation: |det| = area of the parallelogram spanned by the row vectors. If det = 0: singular matrix (irreversible, linearly dependent rows).
Determinant of a 3×3 matrix - Sarrus' rule
Sarrus method: copy the first two columns next to the matrix. Six products: three on the main diagonal (+), three on the anti-diagonal (-). det = a(ei-fh) - b(di-fg) + c(dh-eg). Example: [[1,2,3],[4,5,6],[7,8,9]]. det = 1(45-48) - 2(36-42) + 3(32-35) = 1(-3) - 2(-6) + 3(-3) = -3+12-9 = 0. (Singular matrix!)
Laplace expansion
For an n×n matrix: expand by any row or column. det(A) = Σ (-1)^(i+j) a_ij × M_ij. M_ij = minor (determinant of matrix (n-1)×(n-1) after removing row i and column j). Cofactor = (-1)^(i+j) × M_ij. Select the row/column with the most zeros (minimize calculations). Recursion: 4×4 → four 3×3 → each three 2×2.
Applications of the determinant
Invertibility: det ≠ 0 ⟺ invertible matrix. Systems of equations: Cramer's rule (det A ≠ 0 → unique solution). Geometry: |det| = volume scaling factor for linear transformation. Eigenvalues: det(A - λI) = 0 (characteristic equation). Vector product: det([[i,j,k],[a1,a2,a3],[b1,b2,b3]]).
FAQ
When is the determinant equal to zero?
Two identical rows or columns. One row or column = sum of others (linear relationship). Zero row or column. Singular matrix: a system of equations that has no unique solution or has infinitely many solutions.
How does the determinant change with matrix operations?
Line replacement: det changes the character (-det). Scaling a row by k: det × k. Adding a multiple of a row to another: det unchanged. Transpose: det(A^T) = det(A). Matrix product: det(AB) = det(A)×det(B). Inverse: det(A^{-1}) = 1/det(A).
How to calculate the determinant of 4×4?
Laplace expansion: select the row with zeros. For each element a_ij ≠ 0: calculate minor 3×3 (calculator above). Cofactor: (-1)^(i+j) × minor. Sum: Σ a_ij × cofactor. 4×4 with one zero: 3 operations instead of 4. Gaussian elimination: faster for large matrices.
What are eigenvalues and how are they related to the determinant?
Eigenvalues λ: det(A - λI) = 0 (characteristic equation of degree n). Determinant A = product of all eigenvalues (det A = λ1 × λ2 × ... × λn). Matrix trace = sum of eigenvalues. If 0 is an eigenvalue: det = 0 (singular matrix).
How to check determinant calculations?
Numpy (Python): e.g.linalg.det(array). MATLAB: det(A). Wolfram Alpha: "determinant of [[1,2],[3,4]]". Excel: =MDETERM({1,2,3,4}). Manually: check 2×2 cofactors separately. Red flag: det(A) × det(A^{-1}) ≠ 1 → calculation error.
Related tools: matrix calculator, vector calculator and systems of equations calculator.