Exercises
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Exercises
Solve the following systems of linear equations using the elementary row operations discussed in this section.
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Construct three different augmented matrices that all have the given solutions.
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Explorations
- Given a matrix on which one or more elementary row operation has been performed, is it possible to go back to the original system of linear equations?
- Suppose the system of equations below is inconsitent, i.e. it has no solution. What can you say about the values of $b$ and $c$?
$$3x-4y=a\\bx+cy=d$$
Theory
- The second elementary row operations says that multiplying one of the equations by a (non-zero) constant the new system has the same solution set as the original one. Show that this is true.
- The third elementary row operations says that multiplying one of the equations by a (non-zero) constant and adding it to the another equation gives a new system has the same solution set as the original one. Show that, without loss of generality, you can assume that the constant multiplier is 1.
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Given a matrix on which one or more elementary row operation has been performed, is it possible to go back to the original system of linear equations?
This isn't possible. The row operations preserve the one piece of information that we care about, i.e. the system's solution, but they don't store the specifics of the operations that were used to get to the matrix's current form.
Suppose the system of equation below is inconsitent, i.e. it has no solution. What can you say about the values of $b$ and $c$?
$$3x-4y=a\\bx+cy=d$$Think about this geometrically. To be inconsistent, the lines have to be parallel. Rethink the question in terms of the slopes of the lines.
Suppose the system of equation below is inconsitent, i.e. it has no solution. What can you say about the values of $b$ and $c$?
$$3x-4y=a\\bx+cy=d$$The slope of the first line is $3/4$. For the system to be inconsistent, the second line has to be parallel to the first one, i.e. they have to have the same slopes but different y-intercepts. For that to be the case, we would have to have
$$\frac{-b}{c} = 3/4$$This will be true when $b=-3k$ and $c=4k$, i.e. when the coefficients of the second equation are a constant multiple of the coefficients of the first equation, assuming that this doesn't make $kd = a$. In that case, the system would have infinitely many solutions.
The third elementary row operations says that multiplying one of the equations by a (non-zero) constant and adding it to the another equation gives a new system has the same solution set as the original one. Show that, without loss of generality, you can assume that the constant multiplier is 1.
Think of this as being made up of two steps: first multiply then add. What does the second elementary row operation tell you about that first step?
