Math 1050 - Exam 3 Review

1. Solve the following system of equations in the manner indicated.

(a) By the Method of Substitution
(b) By elimination
(c) By Gaussian elimination (matrices)
(d) By Gauss-Jordan elimination (matrices)

2. Graph each system of linear equations . How many solutions does each system have? Label
and scale your axes.

3. At the movie theater, a bag of popcorn costs $5.00 and a soda costs $3.00. If Darci buys
a total of 8 items and spends a total of $30.00 on just popcorn and soda, how many bags
of popcorn and how many sodas does she buy?

4. Solve the systems of equations using any convenient method .

5. Find the equation of the parabola y = ax^2+bx+c that passes through the points (−1, 2),
(1, 6) and (2, 17).

6. Find the partial fraction decomposition of the following.

7. Write the coefficient matrix and the augmented matrix of the following system.

8. Put the following matrix in row-echelon form and reduced row -echelon form.

9. Perform the matrix operations given

(a) A + B
(b) 4B − A
(c) AB
(d) BA

10. If A is a 3×6 matrix and B is a 6×2 matrix, what dimensions does the product AB have?

11. Solve the matrix equation for X, where

12. Find the inverse of the matrix if it exists. Check your results.

13. (a) Find the inverse of the matrix

(b) Use your answer from part (a) to solve the following system .

14. Evaluate the determinant of the matrix.

(b)
i. Use cofactor expansion
ii. Use the alternative method

15. Use Cramer’s Rule to solve the system of equations.

16. Find the area of the triangle with vertices (0,−2), (2, 0) and (−1, 1).

17. Use a determinant to find whether the following points are collinear: (−1, 3), (0, 5), (1, 7).

18. Decode the following by using the inverse of

Recall that