matrix
Math 1021 Assignment 3
Due at 11:59pm EST on Friday November 12, 2021
• To accommodate students with Letters of accommodation or those who experience technical problems, there
will be no penalty on solution submitted between 12 a.m. on November 13 and 11:59 p.m. on November 14,
2021.
• Solution submitted after 11:59 p.m. on Sunday November 14, 2021 will NOT be graded.
• You must include full solution with justifications. Answers without justification may receive a grade of zero.
• Your solutions should be mathematically and logically coherent, and should be easy for the reader to follow.
• Your solution must include your name and York student ID at the top of the first page.
1. Given n real numbers a1, . . . , an, the Vandermonde matrix is the n × n matrix Vn whose (r, s)-entry is a
s−1
r ,
for r, s = 1, . . . , n. [10 marks]
(a) Show that V4 = LU where
L =
1 0 0 0
1 a2 − a1 0 0
1 a3 − a1 (a3 − a1)(a3 − a2) 0
1 a4 − a1 (a4 − a1)(a4 − a2) (a4 − a1)(a4 − a2)(a4 − a3)
and
U =
1 a1 a
2
1 a
3
1
0 1 a1 + a2 a
2
1 + a1a2 + a
2
2
0 0 1 a1 + a2 + a3
0 0 0 1
.
(b) Use Part (a) to compute the determinant of V4.
(c) Show that if a1, a2, a3 and a4 are distinct real numbers then there exists a unique polynomial of degree
at most three, p(x), satisfying p(a1) = b1, p(a2) = b2, p(a3) = b3 and p(a4) = b4, for any b1, b2, b3, b4 ∈ R.
Read Polynomial Interpolation (Page 163-166) of textbook.
(d) Let
b1 = fourth digit of your York student ID
b2 = fifth digit of your York student ID
b3 = sixth digit of your York student ID
b4 = seventh digit of your York student ID
Find the unique polynomial p(x) of degree at most three satisfying p(2) = b1, p(1) = b2, p(0) = b3 and
p(−1) = b4.
2. In this question, we prove that the area of the triangle T1 equals 1
2
det
1 x1 y1
1 x2 y2
1 x3 y3
. [8 marks]
(x1, y1)
(x2, y2)
(x3, y3)
T1
O
A
B
T2
1
(a) Let T2 be the triangle obtained from T1 by translating the plane from (x1, y1) to (0, 0). Find the coordinates of A and B.
(b) Show that T2 has area 1
2
det ”
(x2 − x1) (x3 − x1)
(y2 − y1) (y3 − y1)
#
.
(c) Show that
det
1 x1 y1
1 x2 y2
1 x3 y3
= det
1 0 0
x1 (x2 − x1) (x3 − x1)
y1 (y2 − y1) (y3 − y1)
= det ”
(x2 − x1) (x3 − x1)
(y2 − y1) (y3 − y1)
#
.
(d) Let
x1 = fourth digit of your York student ID
y1 = fifth digit of your York student ID
x2 = 20 + sixth digit of your York student ID
y2 = seventh digit of your York student ID
x3 = 10 + eighth digit of your York student ID
y3 = 10 + ninth digit of your York student ID
Find the area of the triangle with vertices at (x1, y1), (x2, y2), and (x3, y3).
3. Read Section 3.4 of the textbook An Application to Linear Recurrences
Let
a = 1 + fourth digit of your York student ID
b = 1 + fifth digit of your York student ID
c = 1 + sixth digit of your York student ID
Find a formula for xn of the sequence x0, x1, x2, . . . satisfies the recurrence
xk+2 = (a − 1)xk+1 + axk, for k ≥ 0
with the following initial conditions: [12 marks]
(a) x0 = b and x1 = c;
(b) x0 = −c and x1 = −b.
4. Let δ = 10 + fourth digit of your York student ID and d =
1
δ
. [15 marks]
Let
T =
d 1 − d 0
1 − d d 0
0 0 1
be the transition matrix of the Markov Chain.
(See Section 2.9 of the textbook about Markov Chain.)
(a) Diagonalize T.
(b) Let a =
α
α + β + γ
, b =
β
α + β + γ
and c =
γ
α + β + γ
, where
α = 1+second digit of your York student ID
β = 1+third digit of your York student ID
γ = 1+fourth digit of your York student ID
What is the state vector sn after n transitions if the initial state vector is s0 =
a
b
c
?
(c) Compute s∞ = limn→∞
sn. Show that s∞ is a steady-state for this Markov chain.
2
5. Write a summary, in your own words, of the concepts covered in Sections 3.1-3.4. [5 marks]
(This exercise is designed to help you study for the final exam. Your summary does not have to be long but
should cover the important points we have covered. Any reasonable summary will receive full mark for this
question.)
3
