1 point to lose here!
9
2
11
Name:
Bates
Elkin
Student number:
Testing Weber 134 9 AM - 4PM AND Glover 130 6:30 PM - 9 PM –
YOU MUST BRING your CSU ID
M229 Matrices and Linear Equations
First Examination
Version SAMPLE
(9 pts) 1. Identify each of the following matrices as being in reduced row echelon form (RREF) or not. If not obtain the
equivalent RREF matrix by applying elementary row operations. SHOW ALL STEPS
!
A)
B)
C)
1
0
0
0
0
0
1
0
0
1
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
1
1
1
−2
−2
−2
−1
−2
−1
−2
1
1
2
!
2
1
3
!
1
3
2
(8 pts) 2. Consider the linear system:
Write the augmented matrix withrespect to the
ordering x, y, z, w. Reduce it to reduced
row echelon form. Write the general solution of
the original system as a system of equations.




w−z =1
w+x−y+z =0
y+z =2



3x − y = 1

1 −1 a 0
1 2 1
3. A linear system in x1 , x2 , x3 , x4 , x5 has augmented matrix:  0
0
0 0 1
(where a is an unknown constant)

1 2
0 0 .
−1 5
(4 pts) A) Write the general solution expressing the variables associated with the pivotal columns in terms of the
variables associated with the nonpivotal columns.
(4 pts) B) Write the vector form of the general solution as a linear combination of the distinguished solution
and basic solutions of the associated homogeneous system.
(over)
Version Sample
(6 pts) 4. Let A =
3 2
1 1
1
1






1 −3 2
−1
−1
, B =  −2 1 0  , v =  1  and w =  3 .
1 −2 1
4
−1
Decide if each of the following is defined and if so compute it.
A(w + v) =
vT w =
v − Bw =
(8 pts) 5. A linear function fA transforms the points P, Q, R to P ′ , Q′ , R′ respectively. Write a system of linear equations
that must be satisfied by the entries of the matrix A. Solve for A.
Y
P
Y
P’
1
0
R
1
1
1
1
0
X
1
X
Q
Q’
R’
6. The graph of the polynomial y = a0 + a1 x + a2 x2 + a3 x3 passes through the points (−2, 0), (1, 3), (2, 4)
(2 pts) A) Write the augmented matrix of a linear system whose solution is the coefficients a0 , a1 , a2 , a3 .
(4 pts) B) Solve the system and write the general form of the polynomial of degree at most 3 that passes through
these points.
(2 pts) C) Sketch the data and the polynomial associated with the distinguished solution of the linear system.
c
2008
Department of Mathematics Colorado State University