Part A. (20 points) True or False? Write 'T' if the statement is true and 'F' if the
statement is false. 1. Two fundamental questions about a linear system involve existence and uniqueness.
2. The echelon form. of a matrix is unique.
3. When u and v are nonzero vectors, Span ,uv contains the line through u and origin.
4. The equation Axb is homogeneous if the zero vector is a solution.
5. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly
dependent.
6. If A is a 35 matrix and T is a transformation defined by TAxx, then the domain of
T is 3R .
7. If A and B are nn matrices, and AB BA , then 22A B A B A B .
8. If Axb has a unique solution, and A is 55 , and b is 51 , then Ax0 has a non-
trivial solution.
9. The determinant of a triangular matrix is the product of the entries on the main diagonal.
10. T TTA B A B
11. IfA , B , C are nn matrices and
nABC I
, then 1B AC .
12. If A is an invertible matrix, then 1d e t d e t d e t 1A A I .
13. If a 55 matrix A is invertible, then the columns of A span 5R .
14. The eigenvalues of a matrix are on its main diagonal.
15. If A is a 55 matrix with rank A = 3, then 0 is not an eigenvalue of A .
16. A steady-state vector for a regular stochastic matrix is actually an eigenvector.
17. IfA , B , C are square matrices and AB AC then BC .
18. Let A be a 64 matrix of rank 4. Then the dimension of Nul A , dim Nul A = 0 and A has
4 linearly independent columns.
19. The set
1 0 1
1 , 1 , 0
1 1 1
is an orthonormal basis for 3R .
20. For any matrix A , the product TAA is diagonalizable.
Part B. (80 points) Free response.
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When calculations are asked for, show the details of your work. When interpretations or
explanations are called for, be clear and concise.
21. (10 points) Find all solutions to the system of equations below. Express your
solution(s) as a set.
22. (6 points) The downtown core of Gotham City consists of one-way streets, and the
traffic flow has been measured at each intersection. For the city block shown in Figure
below, the numbers represent the average numbers of vehicles per minute entering and
leaving intersections A, B, C, and D during business hours.
(a) Set up and solve a system of linear equations to find the possible flows
1 2 3 4, , ,f f f f