Determinant of 4×4 Matrix

4×4 determinant: Laplace expansion and Gaussian elimination

With a 4×4 matrix, forget about the Sarrus rule: it works only for 3×3. You have two standard routes instead. The first is Laplace expansion by cofactors. Pick a row or column and compute det = Σ (-1)^(i+j) · aᵢⱼ · Mᵢⱼ, where Mᵢⱼ is the 3×3 minor left after you delete row i and column j. Go for the row or column with the most zeros to cut down the arithmetic. The catch is O(n!) complexity, which gets painful beyond 4×4. The second is Gaussian elimination with LU decomposition: reduce A to upper triangular form, and then det(A) = ± (product of the pivots), where the sign tracks how many row swaps you made. That runs in O(n³) and is what you actually use for big matrices. A few properties worth keeping in mind: det(AB) = det(A)·det(B), det(Aᵀ) = det(A), swapping two rows flips the sign, scaling a row by k scales det by k, and adding a multiple of one row to another leaves det alone. As a quick check, a diagonal 4×4 with entries 2, 3, 1, 5 has det = 2·3·1·5 = 30.

Applications

You meet 4×4 determinants when solving 4×4 linear systems by Cramer's rule, in the 4D Jacobian for a change of variable, and in stability analysis of ODE systems with four state variables, where the sign of the Jacobian's determinant at equilibrium tells the story. They also turn up in 3D computer graphics: a 4×4 homogeneous matrix packs rotation, translation, scaling and projection into one operation across the OpenGL/DirectX pipeline.

FAQ

Can I use the Sarrus rule for 4×4? No. Sarrus is a 3×3-only trick. For 4×4, reach for Laplace expansion or Gaussian elimination.

What does det = 0 mean for a 4×4? The matrix is singular. It has no inverse, its rows and columns are linearly dependent, and the associated linear system either has no solution or infinitely many.

Why are graphics matrices 4×4 if space is 3D? That extra homogeneous coordinate is what lets you write translation as a matrix multiplication and lets perspective projection work at all, neither of which a pure 3×3 matrix can do.