Math 2174 Sp26 - Midterm 1 Sample
1. (10 points) Fill in possible missing values in the following 2 × 3 systems of linear equations to obtain the specified solution type.
Justify your answer in a single sentence.
(Hint: Try and select the simplest values that work, especially with a view to working out question 2 below.)
(b) A UNIQUE SOLUTION
(a) INFINITE SOLUTIONS








□
□
x
+
3
y
=
□
□



x
+
y
=
□ □ 1
x+
□x + □y = 3
□x + □y = □



□x + 2y = □
2y =
Reason:
Reason:
2. (20 points) For EACH of the systems you wrote down in part (a) above
• Compute the row-reduced Echelon form (REF) of the augmented matrix of the system. SHOW ALL ROW OPERATIONS.
• Write out the solution to the system in vector form.


k −4
−1
0


3. (20 points) For what values of k is the matrix  2
k − 1 0 singular?
0
0
1
4. (10 points) Based on your answer to problem 3 (or otherwise), check whether the following set of vectors is LINEARLY DEPENDENT or INDEPENDENT. Show work to support your conclusion
     

0 
−1

 −4
     
 2  , −1 , 0


 0
1 
0

1

5. (20 points) Let P , Q be 3 × 3 matrices, and let S and T be given as S = 0
0
1
1
0

1

1 ,
1

4

T = 3
0
1
1
1

1

1
2
(a) If P T + S = T , find P .
(b) If Q T S = T , find Q .
6. (20 points) True or False? Full credit will be awarded for correct answers, but any work will be considered for partial credit if an
answer is wrong.
(a) If x is a vector, then x−1 always exists .
True
False
(b) Given a vector x, ||x|| = xT x.
True
False
(c) A homogeneous system of 100 equations in 200 variables must have infinitely many solutions.
True
False
(d) A system of 300 equations in 200 variables must have infinitely many solutions.
True
False
True
False
2
(e) If A = AB , then A = B