Overview of Determinants

Applicable ONLY for 3×3 determinants

Steps:

1. Rewrite the first two columns to the right

2. Add products of downward diagonals

3. Subtract products of upward diagonals

System of linear equations: AX = B

Consistent / Inconsistent:

• Consistent → one or more solutions

• Inconsistent → no solution

Matrix method (Martin’s Rule):

If ∣A∣ ≠ 0, X = A⁻¹B ⇒ unique solution

When ∣A∣ = 0:

• (adjA)B = 0 → infinitely many solutions

• (adj⁡A)B ≠ 0 → no solution

Homogeneous system:

AX = 0

• Always consistent

• ∣A∣ ≠ 0 → trivial solution

• ∣A∣ = 0→ infinitely many solutions