Matrix Method for 2×2 Systems

Writing as a matrix equation

System:
ax + by = e
cx + dy = f

Becomes: [[a, b], [c, d]] [[x], [y]] = [[e], [f]]

Or AX = B, so X = A^-1B

Inverse of 2×2 matrix

If A = [[a, b], [c, d]], then A^-1 = (1/(ad-bc)) [[d, -b], [-c, a]]

ad - bc is the determinant. If determinant = 0, no unique solution.

Example:
2x + 3y = 13
5x + 2y = 16
A = [[2, 3], [5, 2]], det = 2×2 - 3×5 = 4-15 = -11
A^-1 = (1/-11) [[2, -3], [-5, 2]]
[[x], [y]] = (1/-11) [[2, -3], [-5, 2]] [[13], [16]] = (1/-11) [[26-48], [-65+32]] = (1/-11) [[-22], [-33]] = [[2], [3]]
So x = 2, y = 3

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