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    Determinants and Cramer's Rule : Row Operations and Effect on Determinant, Computing Determinants and Solving Systems of Equations

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    1. State the elementary row operation being performed and its effect on the determinant
    Start with matrix
    a b
    c d
    a.
    c d
    a b
    b.
    a b
    kc kd
    c.
    a + kc b + kd
    c d
    2. Compute the determinant
    a.
    3 0 4
    2 3 2
    0 5 -1
    b.
    1 3 5
    2 1 1
    3 4 2
    c.
    3 5 -8 4
    0 -2 3 -7
    0 0 1 5
    0 0 0 2
    3. Use row reduction to convert the matrices to echelon for and then compute the determinant of each matrix
    a.
    1 3 0 2
    -2 -5 7 4
    3 5 2 1
    1 1 2 -3
    b.
    1 -1 -3 2
    0 1 5 4
    -1 2 8 5
    3 -1 -2 3
    c.
    1 3 -1 0 -2
    0 2 -4 -1 -6
    -2 -6 2 3 9
    3 7 -3 8 -7
    3 5 5 2 7
    4. Use Cramer's Rule to compute solutions to the following systems of linear equations. (Show your work.)
    a. 5x1 + 7x2 = 3
    2x1 + 4x2 = 1
    b. 4x1 + x2 = 6
    5x1 + 2x2 = 7
    c. 3x1 - 2x2 = 7
    -5x1 + 6x2 = -5
    d. 2x1 + x2 = 7
    -3x1 + x3 = -8
    x2 + 2x3 = -3
    5. Compute the adjugate of the given matrices and use that information to find the inverse of each matrix
    a.
    0 -2 -1
    3 0 0
    -1 1 1
    b.
    3 5 4
    1 0 1
    2 1 1
    c.
    3 6 7
    0 2 1
    2 3 4

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    Determinants and Cramer's Rule, Row Operations and Effect on Determinant, Computing Determinants and Solving Systems of Equations 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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