Math 222 Fall 2022
Linear Algebra with Applications
Final Exam
December 17th, 2022
Name:
ID number:
ˆ Enter your name in the space above.
ˆ Answer the questions in the spaces provided. If you run out of room for an answer,
continue on the back of the page.
ˆ You are allowed one letter-size page with notes. You shouldn’t communicate with
anyone else during the exam.
ˆ Indicate your section below.
□ Section 01 - 9:00 with Brett
□ Section 02 - 11:35 with James
□ Section 03 - 2:30 with Hanwen
Math 222, Final, Fall 2022
Name:
r
1. Let A be a 2 × 2 matrix A = 1 with rows r1 , r2 and det(A) = 5. Find det(B) where
r2
3r1 + 2r2
B=
2r1 + 5r2
Page 1 of 7
Math 222, Final, Fall 2022
Name:
2. A forest contains a population of owls and a population of rats. The rats provide most
of the owl population’s food, so the number of owls depends on the number of rats in
the forest (as well as the number of owls). In particular, the number of owls k + 1 years
from now is given by the recurrence relation:
5
1
Ok+1 = Ok + Rk .
4
8
Similarly, the number of rats one year into the future is a linear function of the current
number of both rats and owls:
Rk+1 = −
3
5
Ok + Rk .
10
4
Suppose
are initially 10 thousand owls and 20 thousand rats in the forest. Let
there
Ok
xk =
describe the populations of owls and rats k years in the future.
Rk
(a) Find the number of owls and the number of rats one year into the future.
(b) Find a matrix A so that xk+1 = Axk .
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Math 222, Final, Fall 2022
Name:
(c) Find the eigenvalues of the matrix A you found above.
(d) Find a diagonalizing matrix P such that P −1 AP is diagonal.
(e) Predict the number of owls and the number of rats in the woods 100 years from
now. Explain your work.
Page 3 of 7
Math 222, Final, Fall 2022
3. Given a matrix
Name:


1
0
1
0 ,
A = −1 1
1 −2 −1
(a) find a basis for its null space null(A).
(b) find a basis for its column space col(A) and the rank of A.
(c) find a 3 × k matrix B and a k × 3 matrix C, so that BC = A, where k is the rank
of A you find in (b).
Page 4 of 7
Math 222, Final, Fall 2022
4. Given a matrix
 
1

(a) find the projection of 2
3
first two columns of A.
Name:


0 1 −2
A = 2 −1 1 
0 0
1
   
1 
 0



2 , −1 , the subspace spanned by the
onto span


0
0
(b) find a QR factorization of A by the Gram-Schmidt process.
Page 5 of 7
Math 222, Final, Fall 2022
Name:
5. Consider the following data: y = a + bx modeling the data below.
x
y
1
0
2 4 5
1 2 3
(a) Find the coefficients a and b for the least squares approximating line y = a + bx
modeling the data.
(b) What do you predict the value of y to be when x = 3.
Page 6 of 7
Math 222, Final, Fall 2022
Name:
6. True or false.
(a) If A and B are n × n and A is not invertible, then AB is not invertible for any
matrix B.
True
False
⃝
⃝
(b) If D is a diagonal n × n matrix, then D is invertible.
True
False
⃝
⃝
(c) If 2 × 2 matrix A has only one eigenvalue, then A is not diagonalizable.
True
False
⃝
⃝
(d) If A is an m × n matrix, then AT A is symmetric.
True
False
⃝
⃝
(e) If A is an m × n matrix, then AT A is invertible.
True
False
⃝
⃝
 

 x

(f) The set y  : y + 2x − z + 1 = 0 forms a subspace.


z
True
False
⃝
⃝
(g) The solution to a linear system Ax = b, where A is m × n and m < n, always exists
for any m × 1 vector b.
True
False
⃝
⃝
(h) A linear system Ax = b, where A is m × n and m < n, will never have a unique
solution.
True
False
⃝
⃝
(i) An n × n matrix having linearly independent columns is always invertible.
True
False
⃝
⃝
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