For a homogeneous system AX = 0, non-trivial solutions exist iff det(A) = 0.

The number of equations in a consistent system with 3 variables and rank 2 is $x$ or more (minimum).

The system AX = B has a unique solution iff:

Using Cramer's rule, the solution to x + y = 5, x - y = 1 is:

(AB)⁻¹ = B⁻¹ A⁻¹ for all invertible matrices A and B.

The inverse of a 2×2 matrix [[a, b], [c, d]] is (1/(ad-bc)) × [[d, -b], [-c, a]]. If ad-bc = 0, the inverse $x$ (exists/does not exist).