Solving systems of linear equations

The inverse-matrix method

• Write the system as AX = B, then X = A⁻¹B, provided det A ≠ 0.
• Example: 2x + 3y = 7 and x − 2y = −7.
  A = [[2, 3], [1, −2]], det A = −4 − 3 = −7, A⁻¹ = (−1/7)[[−2, −3], [−1, 2]].
• X = A⁻¹B = (−1/7)[[−2·7 + (−3)(−7)], [−1·7 + 2(−7)]] = (−1/7)[[7], [−21]] = [[−1], [3]], so x = −1 and y = 3.

Cramer's rule

• For AX = B with det A ≠ 0: x = D_x/D, y = D_y/D, z = D_z/D, where D = det A and D_x is D with the x-column replaced by B.
• Same system: D = −7, D_x = |[[7, 3], [−7, −2]]| = −14 + 21 = 7 → x = 7/(−7) = −1.
• D_y = |[[2, 7], [1, −7]]| = −14 − 7 = −21 → y = (−21)/(−7) = 3.
• Both methods give the same answer; Cramer's rule is faster for a single unknown, the inverse method is better when several right-hand sides share the same A.

Interpreting the determinant