Math 1050-1
Exam 3 Review
Section 7.1: Linear and Nonlinear Systems of Equations
Substitution
Graphing
Review Problems: 8, 18, 64, 70, 74
Section 7.2: Two-Variable Linear Systems
Elimination
Review Problems: 12, 24, 44, 50, 54
Section 7.3: Multivariable Linear Systems
Row-Echelon form
Gaussian elimination
Review Problems: 14, 20, 26, 52
Section 7.4: Partial Fractions
Find partial fraction decompositions
Review Problems: 14, 30, 38, 42
Section 8.1: Matrices and Systems of Equations
Orders of matrices
Gaussian elimination with matrices
Review Problems: 52, 58, 70
Section 8.2: Operations with Matrices
Add and subtract matrices
Multiply matrices by scalars
Matrix multiplication
Review Problems: 8, 24, 28, 30, 44, 52, 61
Section 8.3: The Inverse of a Square Matrix
Verify two matrices are inverses of each other
Use Gaussian Elimination to find the inverse of a square matrix
Use the formula to find the inverse of a 2×2 matrix
Review Problems: 13, 18, 22, 48
Section 8.4: The Determinant of a Square Matrix
Find the determinant of 2×2 matrices
Find the minors and cofactors of a matrix
Find the determinant of square matrices
Review Problems: 8, 16, 40, 50
Section 8.5: Applications of Matrices and Determinants
Cramer’s Rule
Area of a triangle
Collinear Points
Equations of lines using determinants
Cryptography
Review Problems: 2, 8, 16, 34, 42
Notes:
1. I will not give you the partial fraction decomposition factors. You will need to know how to set up the
decomposition.
2. You might find it useful to memorize the formula for finding the inverse of a 2×2 matrix.
3. I will not give you the formulas associated to finding determinants, minors, or cofactors.
4. I will give you any equations you might need for section 8.5. For example, possible problems from
section 8.5 could be:
(a) Recall Cramer’s Rule: If a system of n linear equations in n variables has a coefficient matrix A
with nonzero determinant |A|, the solution of the system is
x1 =
|A1 |
|A2 |
|An |
, x2 =
, · · · , xn =
|A|
|A|
|A|
where the ith column of Ai is the column of constants in the system of equations. If the determinant of the coefficient matrix is zero, the system has either no solution or infinitely many
solutions.
Solve the following system of equations using Cramer’s Rule:
4x − 2y + 3z = −2
2x + 2y + 5z = 16
8x − 5y − 2z = 4
(b) Recall the area of a triangle with vertices (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) is
x y1 1 1 1
Area = ± x2 y2 1 2
x3 y3 1 where the symbol ± indicates that the appropriate sign should be chosen to yield a positive area.
What is the area of the triangle with vertices at (0, 0), (5, −2), and (4, 5)?
(c) The test for collinear points says three
only if
points (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) are collinear if and
x1
x2
x3
y1
y2
y3
1
1
1
= 0.
Are the points (0, 1), (4, −2), and (−2, 25 ) collinear?
(d) An equation of the line passing through
the distinct points (x1 , y1 ) and (x2 , y2 ) is given by
x y 1 x1 y1 1 = 0.
x2 y2 1 Using this method, what is the equation of the line passing through the points (10, 7) and (−2, −7)?