2×2 and 3×3 matrices: determinant, inverse, system
det(2×2) = ad − bc
The matrix calculus of the syllabus, in 2×2 and 3×3: determinant and trace, inverse by the cofactor matrix, transpose, product in both orders because AB ≠ BA is the lesson of the chapter, and solving the system A·x = b. A singular matrix is detected by a threshold relative to its scale, and the tool then says what is true: no inverse, and a system with zero or infinitely many solutions, never infinities served as results.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasdet(2×2) = ad − bc
det(2×2) = ad − bc
det(3×3): cofactor expansion along the first row
A⁻¹ = cofactor(A)ᵀ / det(A)
A·x = b ⇒ x = A⁻¹·b
The determinant decides everything: non-zero, the matrix is invertible and the system has a unique solution; zero, the matrix flattens a direction, the inverse does not exist and the system loses its uniqueness. The same number governs all three operations of the tool.
- Determinant
- · the area factor (2×2) or volume factor (3×3) of the transformation; its sign says whether it flips orientation.
- Trace
- · the sum of the diagonal terms.
- Singular
- · of zero determinant: the transformation flattens space, two distinct points can merge.
- Cofactor matrix
- · the matrix of cofactors; its transpose divided by the determinant gives the inverse.
Validity domainThe cofactor formulas are exact and perfectly suited to sizes 2 and 3; beyond, their cost explodes and numerical methods (Gaussian elimination, decompositions) take over, with other precautions.
The cofactor formulas are exact and perfectly suited to sizes 2 and 3; beyond, their cost explodes and numerical methods (Gaussian elimination, decompositions) take over, with other precautions. The singularity threshold is relative to the matrix’s scale, a determinant of 10⁻⁶ being zero for a matrix in the millions and clear-cut for one in the thousandths; at the exact border, floating-point arithmetic cannot tell a true zero from a near-zero, and the tool chooses to warn rather than serve an inverse with giant coefficients.
Reading the resultThe product is computed in both orders, AB and BA, because their difference is the most important lesson of the chapter: the order of transformations matters, rotating then stretching is not stretching then rotating.
The product is computed in both orders, AB and BA, because their difference is the most important lesson of the chapter: the order of transformations matters, rotating then stretching is not stretching then rotating. When the two coincide, the tool says so as a special case. For a singular system, the honest answer is not a vector: it is the alternative, no solution if the equations contradict each other, infinitely many if they repeat.