Vectors in 2-Space and 3-Space : Properties of the determinant function
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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 ,
(a cos + b sin)2  a2 + b2
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Vectors in 2-Space and 3-Space : Properties of the determinant function
2. Verify that det(AB) = det(A) det(B) for
A= 2 1 0 and B = 1 -1 3
340 7 12
002 5 01
Is det(A+B) = det(A) + det(B) ?
5. Let A = a b c
def
ghi
Assuming that det(A) = -7, find
b) det(A-1)
e) det a g d
bhe
ci 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).
C(x3, y3) •
· B(x2, y2)
•
A(x1, y1)
D E F
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
w
f) 1 w
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
2 2 2 2
v1 + v2 + … + vr = v1 + v2 + … + vr
26. Use the Cauchy-Schwarz inequality to prove that for all real values of a, b, and θ,
(a cosθ + b sinθ)2 ≤ a2 + b2
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