Exercise 1 shows that we can calculate a 3 ×3 determinant by “expanding” along any row and...

Exercise 1 shows that we can calculate a 3 ×3 determinant by “expanding” along any row and remembering the pattern of signs

associated with the terms of the expansion. For instance,

Use this observation to provide a quick proof of the following.

(a)

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(b)

Exercise 1

(a) Use Exercises 2, 3, and 4 (or direct calculation) to prove that

 

for any three vectors a, b, and c in ℝ3.

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(b) Show that

 

for any three vectors a, b, and c in ℝ3.

Exercises 2–4

2. Prove that .

3. Prove that

4. Prove that