# Vectors in 2-Space and 3-Space : Properties of the determinant function

Please see the attached file for the fully formatted problems.

2. Verify that det(AB) = det(A) det(B) for

A = 2 1 0 and B = 1 -1 3

3 4 0 7 1 2

0 0 2 5 0 1

Is det(A+B) = det(A) + det(B) ?

5. Let A = a b c

d e f

g h i

Assuming that det(A) = -7, find

b) det(A-1)

e) det a g d

b h e

c i f

9. Prove the identity without evaluation the determinants.

a1 + b1 a1 - b1 c1 a1 b1 c1

a2 + b2 a2 - b2 c2 = -2 a2 b2 c2

a3 + b3 a3 - b3 c3 a3 b3 c3

16. Let A and B be n x n matrices. Show that if A is invertible,

then det(B) = det(A-1BA)

18. Prove that a square matrix A is invertible if and only if ATA is invertible.

10. a) In the accompanying figure, the area of the triangle ABC can be expressed as

area ABC = ½ x1 y1 1

x2 y2 1

x3 y3 1

Note: In the derivation of this formula, the vertices are labeled such that the triangle is traced counterclockwise proceeding from (x1, y1) to (x2, y2) to (x3, y3). For a clockwise orientation, the determinant above yields the negative of the area.

b) Use the result in (a) to find the area of the triangle

with vertices (3,3), (4,0), (-2, -1).

Figure Ex-10

Euclidean Vector Spaces: Euclidean n-Space

6. Let u = (4, 1, 2, 3), v = (0, 3, 8, -2), and w = (3, 1, 2, 2). Evaluate each expression.

a) u + v

b) u + v

c) -2u + 2 u

d) 3u - 5v + w

e) 1

w

f) 1

w

16. Find two vectors of norm 1 that are orthogonal to the tree vectors u = (2, 1, -4),

v = (-1, -1, 2, 2), and w = (3, 2, 5, 4).

20. Find u ? v given that u + v = 1 and u - v = 5

24. Prove the following generalization of Theorem 4.1.7. If v1, v2, ..., vr are pairwise orthogonal vectors in Rn, then

v1 + v2 + ... + vr 2 = v1 2 + v2 2 + ... + vr 2

26. Use the Cauchy-Schwarz inequality to prove that for all real values of a, b, and theta,

(a cos(theta) + b sin(theta))^2 =< a^2 + b^2

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#### Solution Summary

Vectors in 2-Space and 3-Space and properties of the determinant function are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.