MAT223 – Linear Algebra I – Winter 2023
Tutorial – Week 2
Suggested Problems
1. Consider the following system of linear equations, which we refer to throughout this question.
2x1 + x2 − 3x3 = 0
x1 + x2 + x3 = 1
(a) What are the correct m and n such that it is an m × n system?
(b) What is the augmented matrix of the system?
(c) What are the correct m and n such that the augmented matrix from part (b) is m × n? Think about
why this answer is different from your answer to part (a)!
(d) What is the REF of your augmented matrix? What is the RREF?
(e) Find the solutions to this system using the RREF.
(f) Finally, take the matrix in REF (not RREF), translate that back into a system of equations, and find
the solutions. Was this easier or harder than what you did with the RREF?
2. Give three examples of systems of linear equations, each with two equations in two variables, that have:
(a) Exactly one solution.
(b) Infinitely many solutions.
(c) No solutions.
Think about how this relates to the ways that two lines in the plane can intersect.
3. Try to think about these questions geometrically. If you think they’re not possible, explain why. If you think
they are possible, sketch a picture of what such a system would look like.
(a) Is it possible for there to be exactly two solutions to a system of two equations in two variables?
(b) Is it possible for there to be exactly two solutions to a system of any number of equations in two
variables?
(c) Is it possible for there to be no solutions to a system of two equations in three variables?
(d) Is it possible for the set of solutions to a system of five equations in three variables to form a line?
4. Think about a 7 × 23 system of linear equations. What are the possible ranks of such a system? What are
the possible numbers of free variables/parameters such a system can have? What’s the relationship between
the answers to those two questions?
5. True or false?
For each part, determine if you believe the statement is true or false, and justify your answers as appropriate.
(a) If an m × n system has more variables than equations, then it must have infinitely many solutions.
(b) If an m × n consistent system has more variables than equations, then it must have infinitely many
solutions.
(c) If the augmented matrix of a consistent system is m × n, where n > m, then the system must have
infinitely manys solutions.
(d) If an m × n system has a coefficient matrix with rank r and an augmented matrix with rank r + 1, then
the system is inconsistent.
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