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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
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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