3x3 Matrix Calculator
Calculate determinant using Sarrus rule, transpose, and inverse of any 3x3 matrix.
Returns all results for linear algebra and 3D graphics transformations.
A 3×3 matrix is a rectangular array of 9 numbers arranged in 3 rows and 3 columns. Matrices are used to represent and solve systems of equations, perform geometric transformations, and describe many physics and engineering systems.
Determinant of a 3×3 matrix: For matrix M with elements arranged as: |a b c| |d e f| |g h i|
det(M) = a(ei − fh) − b(di − fg) + c(dh − eg)
Worked example: |2 3 1| |4 0 2| |1 5 3|
det = 2(0×3 − 2×5) − 3(4×3 − 2×1) + 1(4×5 − 0×1) = 2(0 − 10) − 3(12 − 2) + 1(20 − 0) = 2(−10) − 3(10) + 20 = −20 − 30 + 20 = −30
If the determinant is zero, the matrix is “singular” — it has no inverse and represents a system with no unique solution.
Matrix multiplication: Two matrices can be multiplied if the number of columns in the first equals the number of rows in the second. For two 3×3 matrices A and B, each entry Cᵢⱼ = row i of A dotted with column j of B.
Inverse of a 3×3 matrix: M⁻¹ = (1/det(M)) × adjugate(M) Only exists when det(M) ≠ 0.
Transformations in 3D space:
- Rotation around z-axis by θ: |cos θ −sin θ 0| |sin θ cos θ 0| |0 0 1|
Applications:
- Solving systems of 3 linear equations simultaneously
- 3D computer graphics (rotation, scaling, projection)
- Physics: inertia tensors, stress matrices
- Statistics: covariance matrices
- Economics: input-output models (Leontief matrices)