Linear Algebra

320 Chapter Four. DeterminantsIIIOther Formulas(This section is optional. Later sections do not depend on this material.)Determinants are a fount of interesting and amusing formulas. Here is onethat is often seen in calculus classes and used to compute determinants by hand.III.1 Laplace’s Expansion1.1 Example In this permutation expansion∣ ∣ ∣ t 1,1 t 1,2 t 1,3∣∣∣∣∣∣∣∣∣∣ 1 0 0∣∣∣∣∣ 1 0 0t 2,1 t 2,2 t 2,3 = t 1,1 t 2,2 t 3,3 0 1 0∣t 3,1 t 3,2 t 3,3 0 0 1∣ + t 1,1t 2,3 t 3,2 0 0 10 1 0∣∣ ∣ ∣∣∣∣∣ 0 1 0∣∣∣∣∣ 0 1 0+ t 1,2 t 2,1 t 3,3 1 0 00 0 1∣ + t 1,2t 2,3 t 3,1 0 0 11 0 0∣∣ ∣ ∣∣∣∣∣ 0 0 1∣∣∣∣∣ 0 0 1+ t 1,3 t 2,1 t 3,2 1 0 00 1 0∣ + t 1,3t 2,2 t 3,1 0 1 01 0 0∣we can, for instance, factor out the entries from the first row⎡ ∣ ∣ ⎤∣∣∣∣∣ 1 0 0∣∣∣∣∣ 1 0 0= t 1,1 · ⎣t 2,2 t 3,3 0 1 00 0 1∣ + t 2,3t 3,2 0 0 1⎦0 1 0∣⎡ ∣ ∣ ⎤∣∣∣∣∣ 0 1 0∣∣∣∣∣ 0 1 0+ t 1,2 · ⎣t 2,1 t 3,3 1 0 00 0 1∣ + t 2,3t 3,1 0 0 1⎦1 0 0∣⎡ ∣ ∣ ⎤∣∣∣∣∣ 0 0 1∣∣∣∣∣ 0 0 1+ t 1,3 · ⎣t 2,1 t 3,2 1 0 00 1 0∣ + t 2,2t 3,1 0 1 0⎦1 0 0∣and swap rows in the permutation matrices to get this.⎡ ∣ ∣ ⎤∣∣∣∣∣ 1 0 0∣∣∣∣∣ 1 0 0= t 1,1 · ⎣t 2,2 t 3,3 0 1 00 0 1∣ + t 2,3t 3,2 0 0 1⎦0 1 0∣⎡ ∣ ∣ ⎤∣∣∣∣∣ 1 0 0∣∣∣∣∣ 1 0 0− t 1,2 · ⎣t 2,1 t 3,3 0 1 00 0 1∣ + t 2,3t 3,1 0 0 1⎦0 1 0∣⎡ ∣ ∣ ⎤∣∣∣∣∣ 1 0 0∣∣∣∣∣ 1 0 0+ t 1,3 · ⎣t 2,1 t 3,2 0 1 00 0 1∣ + t 2,2t 3,1 0 0 1⎦0 1 0∣