1. More on Determinants; Area and Volume (10/19/05) How is det(A) affected by row operations? Replacing one row with a multiple of another row has no effect. Interchanging two rows causes det(A) to change signs. Multiplying a row by k (i.e., scaling) also multiplies det(A) by k. Hence a strategy to “ease the pain” of calculating determinants is to do some row operations on the given matrix.

2. A simple formula Since any square matrix A can be put into echelon (i.e., upper triangular) form using only interchanges and row replacement (no scaling), we see that det(A) = (-1) r (the product of the pivots of its echelon form) where r is the number of interchanges which were needed.

3. Relationship to invertibility and matrix multiplication The formula just given show that A square matrix A is invertible if and only if det(A)  0. Moreover, the determinant is multiplicative, i.e., if A and B are both n by n square matrices, then det(A B) = det(A) det(B)

4. Area and Volume Theorem. If A is a 2 by 2 matrix, the area of the parallelogram determined by the columns of A is |det(A)|. If A is 3 by 3, the volume of its parallelepiped is also |det(A)|. The proof follows from the fact that when a rectangle is “sheared” to a parallelogram, the area is unchanged.

5. Expansion or contraction by a linear transformation A linear transformation T from R 2 to R 2 or from R 3 to R 3 causes expansion or contraction by an amount measured by |det(A)| (where A is the standard matrix of T). Specifically, if S is a set in R 2 with finite area, then area(T (S )) = |det(A)| area(S ). Likewise in R 3 with volumes.

6. Assignment for Wednesday (!) Yes, that’s right. Friday is “Study Day” (so don’t forget to study!) and next Monday we will have a lab on an application of linear algebra. For Wednesday: Read Section 3.2 and Section 3.3 from the bottom of page 204 on. In 3.2 do #1 – 11 odd, 15 – 19 odd, 25, 27 and 31. In 3.3 do 19 – 27 odd.