Linear Algebra MAT208
Prof. Porter Exam #1
NAME:________________________________
1. Given the system of equations:
x1 2 x2
 x1
x2
x2
3x1
x3
 x4  7
x4  3
2 x3
x3
x4  5
x4  8
a) Give the coefficient matrix for the system.
b) Give the augmented matrix that you would use to
solve the system.
c) Give the row reduced echelon form of the system to
find the solution
d) What is the solution?
e) What are the pivots?
2. Set up and solve the system of equations used to find the equation of the parabola going thru
the points (0,4),(1,8),(2,4)
Now add the point (3, 0). What would the new system of equations look like?
Is the system with the new point consistent?
Write your system of equations as a linear combination of column vectors in R 4.
How many pivots does your system have?
3. Give any non-singular, non diagonal 3 x 3 matrix A and find the determinant.
See if your A matrix has an inverse. If so, how would you use the inverse to solve the equation
Ax = [1 2 3]T ?
Do all matrices of this type have an inverse?
4. Find (A+B)C if given the following:
1 2 
 0 3
1 
A
B
C=   D=CT


 2 4
 3 0
 2
Find ||AC + BC||
Find CD
x
5. Give a geometric interpretation for : W= {X : X=  y  , x  y, z  0 }
 
 z 
What do you need to show this is a subspace?
Determine if the above subset is subspace. Show all work.
6. Find the null space for the Matrices:
1 2 0 
A  0 0 1 
0 0 0 
1 2 0 
B  0 0 1 
 3 0 0 
1
0
C
0

0
2 0 7
1 2 5 
0 0 0

0 0 0
Identify the pivot columns
Identify the free variables
Find the special solutions for the above matrices.