DETERMINANTS AND EIGENVALUES 1. Introduction Gauss ...

3. SOME IMPORTANT PROPERTIES OF DETERMINANTS 87(Do you see a quick way to compute that?) Hence,⎡det ⎣ 1 0 2⎤5 1 2⎦3 0 6x 1 == 0 2 2 =0⎡det ⎣ 1 1 2⎤1 5 2⎦2 3 6x 2 == 8 2 2 =4⎡det ⎣ 1 0 1⎤1 1 5⎦2 0 3x 3 == 1 2 2 .You should try to do this by Gauss-Jordan reduction.Cramer’s rule is not too useful for solving specific numerical systems of equations.The only practical method for calculating the needed determinants for n large is touse row (and possibly column) operations. It is usually easier to use row operationsto solve the system without resorting to determinants. However, if the system hasnon-numeric symbolic coefficients, Cramer’s rule is sometimes useful. Also, it isoften valuable as a theoretical tool.Cramer’s rule is related to expansion in minors. You can find further discussionof it and proofs in Section 5.4 and 5.5 of Introduction to Linear Algebra by Johnson,Riess, and Arnold. (See also Section 4.5 of Applied Linear Algebra by Noble andDaniel.)Appendix. Some Proofs. Here are the proofs of two important theoremsstated in this section.The Product Rule. det(AB) = (det A)(det B).Proof. First assume that A is non-singular. Then there is a sequence of rowoperations which reduces A to the identityA → A 1 → A 2 → ...→A k =I.Associated with each of these operations will be a multiplier c i which will dependon the particular operation, anddet A = c 1 det A 1 = c 1 c 2 det A 2 = ···=c 1 c 2 ...c k det A k = c 1 c 2 ...c ksince A k = I and det I = 1. Now apply exactly these row operations to the productABAB → A 1 B → A 2 B → ...→A k B =IB = B.The same multipliers contribute factors at each stage, anddet AB = c 1 det A 1 B = c 1 c 2 det A 2 B = ···=c 1 c 2 ...c} {{ k det B = det A det B.}det A