2025 WTW211
Assignment 01
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1. For each of the statements below, State whether it is true or false, and Motivate your answer
with either a proof, a reference to a definition or theorem, or a counterexample.
For each of the statements, where appropriate, assume that A, B and C are m × n matrices,
x, b, d ∈ Rn be column vectors, where m > 1, n > 1 are integers.
(a) If the systems Ax = b and Cx = d are equivalent and Au = b for some vector u, then we
must have that Cu = b.
(b) Let A, B and C be matrices. If A and C are row equivalent, and B and C are row
equivalent, then A and BC are row equivalent.
(c) If A and C are row equivalent matrices and Cu = b for some vector u then Au = b.

 x2 − 4x3 + x4 = 2
2x1 + x2 + x3 = 1
2. Given a linear system

x1 + x2 − 2x3 = 1
(a) Write the augmented matrix [A|b] of this system.
(b) Use Gauss-Jordan elimination to obtain the reduced row echelon form of [A|b].
(c) Use the ranks of A and [A|b] to confirm that the system is consistent.
(d) After identifying fixed variables and free variables, obtain the solution set of this system.


1+t
 0 

3. Use definition of a solution of a system of linear equations, to determine if v := 
 0  for
t

=1
 x1 + 2x2 − x4
2x2 + x3
=0 .
all t ∈ R is a solution to the system

−x1 + 3x2 + 2x3 + x4 = −1
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