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Determinants, inverse matrices and solving systems sheet
Essay frame

Given A = [1 2 3; 0 1 4; 5 6 0], find A⁻¹ by the cofactor method and use it to solve x + 2y + 3z = 14, y + 4z = 14, 5x + 6y = 17.

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  1. 1 · Find the determinant. |A| = 1(0 − 24) − 2(0 − 20) + 3(0 − 5) = 1 ≠ 0, so A⁻¹ exists.
  2. 2 · Matrix of minors and cofactors. Apply the sign pattern [+ − +; − + −; + − +]: cofactors [−24 20 −5; 18 −15 4; 5 −4 1].
  3. 3 · Adjoint. Transpose: adj(A) = [−24 18 5; 20 −15 −4; −5 4 1].
  4. 4 · Inverse. A⁻¹ = adj(A) ÷ 1 = [−24 18 5; 20 −15 −4; −5 4 1].
  5. 5 · Solve X = A⁻¹B. With B = [14; 14; 17]: x = 1, y = 2, z = 3.
  6. 6 · Check. Row 2: 2 + 12 = 14 ✓; row 3: 5 + 12 = 17 ✓.
  7. 7 · Business link. Keep A⁻¹: next month's targets need only one new multiplication.