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(MATH111)[2009](f)final~cs_ysx^_10433.pdf
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Final Exam for Math 111 (L1), Dec. 19, 2009
Name: ................., ID: ................., Score: .................
There are 45 questions, each of which has only one correct statement. Please enter your answer into the answer tables. There is one answer table on each side of this sheet. For a matrix A, we use r(A), AT , Col A, Null A and Row A to denote the rank, the transpose, the column space, null space, and row space of A respectively.

1111
..
For questions (1) -(6) below, we assume that A =0 1 0 1. 0202
(1)
A linear system having A as its augmented-coe.cient matrix has a) no solution, b) a general solution parameterized by one free parameter, c) a general solution parameterized by two free parameters, d) a general solution parameterized by three free parameters.

(2)
Matrix equation A.x = .0 has a) no solution, b) a general solution parameterized by one free parameter, c) a general solution parameterized by two free parameters, d) a general solution parameterized by three free parameters.

(3)
The column space of A is a a) two-dimensional subspace of R3 , b) two-dimensional subspace of R4 , c) one-dimensional subspace of R3 , d) one-dimensional subspace of R4 .

(4)
The null space of A is a a) two-dimensional subspace of R3 , b) two-dimensional subspace of R4 , c) one-dimensional subspace of R3 , d) one-dimensional subspace of R4 .

(5)
The column space of AT is a a) two-dimensional subspace of R3 , b) two-dimensional subspace of R4 , c) one-dimensional subspace of R3 , d) one-dimensional subspace of R4 .

(6)
The null space of AT is a a) two-dimensional subspace of R3 , b) two-dimensional subspace of R4 , c) one-dimensional subspace of R3 , d) one-dimensional subspace of R4 .


-
1 1 1
For questions (7) -(15) below, we assume that A = .0 .1 0..
0 2 1
(7) The determinant of A is a) 1, b) 2, c) .1, d) .2.
(8) The determinant of A2009 is. a) 1, b) 22009 ,. . c) .1,. d) .22009 .. . . .
1 0 0 1 .1 1 1 .1 .1 1 1 1
(9) The inverse of A is a) .1 .1 2., b) .0 .1 0., c) .0 .1 0 ., d) .0 1 0.
1 0 1 0 2 1 0 2 1 0 0 1
(10) The determinant of A.2009 is a) 1, b) 22009 , c) .1, d) .22009 .
(11) The set of eigenvalues of A is a) {0, 1, 2}, (12) The number of eigenspaces of A is a) 1, b) {.1, 0, 1}, c) {0, 1}, b) 2, c) 3, d) 4. d) {1, .1}.
(13) The sum of the dimensions of the eigenspaces of A is a) 1, b) 2, c) 3, d) 4.

(14)
a) A is diagonalizable, b) A is not diagonalizable, c) A is a symmetric matrix, d) none of the above is true.

(15)
a) AAT is diagonalizble, b) AT is digonalizable, c) AAT is not diagonalizable, d) A.1 is diagonal-izable.

(16)
Let A and B be two row equivalent matrices. Suppose that B is in echelon form