3x3 Matrix Calculator

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 it represents a system with no unique solution.

Why zero needs a tolerance. Take a matrix whose second row is 0.1 times the first. On paper the determinant is exactly zero. In binary, 0.1 cannot be stored exactly, so the arithmetic lands near −5×10⁻¹⁸ rather than on 0. Divide by that and you get an “inverse” with entries in the quadrillions, printed under a determinant already rounded to 0. The check here is whether the determinant is negligible next to the size of the terms that built it, not whether it lands on zero exactly.

Inverse of a 3×3 matrix: M⁻¹ = (1/det(M)) × adjugate(M) Only exists when det(M) ≠ 0. Every inverse below is multiplied back against the original, so you can see the identity matrix appear rather than take the result on trust.

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)