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    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 theta,

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

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    https://brainmass.com/math/vector-calculus/properties-determinant-function-118206

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    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.

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