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)
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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