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

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

Please see the attached file for the fully formatted problems.

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#### Solution Summary

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.