MATH 285, HONORS MULTIVARIABLE CALCULUS, FALL 1999 Problem Set 3

Worksheet: Determinants in 3 dimensions (in class)

Let a⃗ = ⟨a_{1}, a_{2}, a_{3}⟩, b⃗ = ⟨b_{1}, b_{2}, b_{3}⟩, and c⃗ = ⟨c_{1}, c_{2}, c_{3}⟩ be three 3-dimensional vectors. The determinant of a⃗, b⃗, and c⃗ is written as det(a⃗, b⃗, c⃗) or

| \begin{matrix}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} | .

It is defined as

det(a⃗, b⃗, c⃗) = a_{1} | \begin{matrix}b_{2} & b_{3} \\ c_{2} & c_{3} \end{matrix} | − a_{2} | \begin{matrix}b_{1} & b_{3} \\ c_{1} & c_{3} \end{matrix} | + a_{3} | \begin{matrix}b_{1} & b_{2} \\ c_{1} & c_{2} \end{matrix} | .

Problem 1 Calculate

| \begin{matrix}1 & 3 & 3 \\ 0 & 4 & −2 \\ 2 & 2 & 1 \end{matrix} | .

Problem 2 Write down a formula for det(a⃗, b⃗, c⃗) that does not use 2-dimensional determinants.

Problem 3 Prove that

| \begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & b_{2} & b_{3} \\ 0 & 0 & c_{3} \end{matrix} | = a_{1}b_{2}c_{3}.

Problem 4 Prove that the determinant satisfies the following properties:

1. det(a⃗, b⃗, c⃗) = det(b⃗, c⃗, a⃗) = det(c⃗, a⃗, b⃗), i.e., when rotating the rows, the determinant does not change.

2. det(a⃗, b⃗, c⃗) = − det(b⃗, a⃗, c⃗), i.e., when swapping two rows, the determinant changes signs.

3. det(λa⃗, b⃗, c⃗) = det(a⃗, λb⃗, c⃗) = det(a⃗, b⃗, λc⃗) = λ det(a⃗, b⃗, c⃗), where λ is a scalar.

4. det(a⃗ + a′, b⃗, c⃗) = det(a⃗, b⃗, c⃗) + det(a′, b⃗, c⃗).

Problem 5 Conclude from the above properties that the determinant does not change when a multiple of one row is added to another row. In symbols, det(a⃗, b⃗, c⃗) = det(a⃗, b⃗ + λa⃗, c⃗). Use this fact, together with Problem 3, to find a quicker way of calculating the determinant in Problem 1. (Hint: use a form of Gaussian elimination).

Homework Problems (due Tuesday 10/5)

13.1 #12, 16, 40; 13.2 #30, 32, 34 (see Example 5, p. 828); 13.3 #58, 61, 62;