Written Assignment 2
Please submit solutions to the following questions to the dropbox labelled Math 1025N near the elevator
on 5th oor N Ross building (due: 11:59 p.m. on March 1). Please write down your full name and
student ID on your solution sheets (A4 size papers).
1. (Question contributed by a student) Prove the following statement:
By elementary row operations, a matrix may be transformed into two di erent row-echelon forms which
are not in reduced row-echelon form. (Hint: You may use the matrix below.)
2
4
3 1 5
0 1 10
2 1 0
3
5
2. Show that the system
8>
x + 2y z = a
2x + y + 3z = b
x 4y + 9z = c
is inconsistent unless c = 2b 3a.
3. We will nd a quadratic polynomial a + bx + cx2 such that the graph of y = a + bx + cx2 contains each
of the points ( 1;6), (2;0) and (3;2).
(a) Write down a system of linear equations with variable a, b and c.
(b) Solve the system using Gauss-Jordan elimination and nd the coe cients of the quadratic polyno-
mial.
4. In each of the following, nd (if possible) conditions on a and b such that the system has no solution,
unique solution, and in nitely solutions.
(a) (
x 2y = 1
ax + by = 5
(b) (
x by = 1
x + ay = 3
5. Solve the following system of equations by Gauss-Jordan elimination.
8
>:
x1 + 3x2 2x3 = 3
2x1 + 6x2 2x3 + 4x4 = 18
x2 + x3 + 3x4 = 10
6. Can two equations with three variables have a unique solution? Give two reasons for your answer by...
(a) examining the possible positions of planes in space.
(b) using Theorem 1.36 (Rank and Solutions to a Consistent System of Equations) of the textbook.
7. Find the matrix A which satis es
AT
3 1
0 1
T
= 2A + 3
2 1
4 1
:
8. Find the inverse of the matrix 2
4
1 1 1
0 1 2
0 0 k
3
5
in terms of k. Determine all value(s) of k for which the above matrix is not invertible.
9. Show that
(a) If A is invertible, A26= 0. (Hint: Assume that there exist an invertible matrix A such that A2 = 0,
and then show that such an assumption leads to a contradiction.)
(b) If A2 is invertible, then A is invertible.
10. Let
A =
1 4
2 2
:
(a) Express A 1 as a product of elementary matrices.
(b) Express A as a product of elementary matrices.
This is the end of Written Assignment 2. (Final update: Feb 16)
*Not every question will be graded.