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\)
Discovery guide 2.1 Discovery guide
Reminder.
The elementary row operations are
swap rows;
multiply a row by a non-zero constant; and
add a multiple of one row to another.
Discovery 2.1 .
Consider the following system.
\begin{equation*}
\left\{\begin{array}{rcrcrcr}
2x \amp \amp \amp - \amp 2z \amp = \amp 4, \\
x \amp - \amp y \amp \amp \amp = \amp 3,\\
4x \amp - \amp 2y \amp - \amp 3z \amp = \amp 7.
\end{array}\right.
\end{equation*}
(a) Convert to an augmented matrix.
(b) Via elementary row operations, obtain a “leading \(1\) ” in the first entry of the first row (maybe swap some rows?), then use it to eliminate all other entries in the first column.
(c) Obtain a leading \(1\) in the second entry of the second row (do not use/alter the first row!), then use it to eliminate all other entries in the second column (yes, you can now alter the first row).
(d) Obtain a leading \(1\) in the third entry of the third row (do not use/alter first or second rows!), then use it to eliminate all other entries in the third column.
(e) Turn the final augmented matrix back into a system and solve it.
Discovery 2.2 .
Consider the following system.
\begin{equation*}
\left\{\begin{array}{rcrcrcr}
3x \amp + \amp 6y \amp + \amp 5z \amp = \amp -9,\\
2x \amp + \amp 4y \amp + \amp 3z \amp = \amp -5,\\
3x \amp + \amp 6y \amp + \amp 6z \amp = \amp -12.
\end{array}\right.
\end{equation*}
(a) Convert to an augmented matrix.
(b) Via elementary row operations, obtain a leading \(1\) in the first entry of the first row (maybe combine first two rows somehow?), then use it to eliminate all other entries in the first column.
(c) Is it possible to obtain a leading \(1\) in the second entry of the second row?
(d) Obtain a leading \(1\) in third entry of the second row (do not use/alter the first row!), then use it to eliminate all other entries in the third column.
(e) Assign a parameter to every variable whose column does not contain a leading one. Turn the final augmented matrix back into a system and solve it in terms of your parameter(s).
Discovery 2.3 .
Consider the following system.
\begin{equation*}
\left\{\begin{array}{rcrcrcr}
x \amp + \amp 2y \amp + \amp z \amp = \amp 2,\\
2x \amp + \amp 5y \amp + \amp 2z \amp = \amp -3,\\
2x \amp + \amp 4y \amp + \amp 2z \amp = \amp -1.
\end{array}\right.
\end{equation*}
(a) Convert to an augmented matrix.
(b) Use the leading \(1\) in first entry of the first row to eliminate all other entries in the first column.
(c) Convert the new third row back into an equation. What does this mean about the system?
Discovery 2.4 .
Consider the following system. Notice that the “equals” column is all zeros. Such a system is called homogeneous .
\begin{equation*}
\left\{\begin{array}{rcrcrcrcrcr}
3x_1 \amp + \amp 6x_2 \amp - \amp 8x_3 \amp + \amp 13x_4 \amp = \amp 0,\\
x_1 \amp + \amp 2x_2 \amp - \amp 2x_3 \amp + \amp 3x_4 \amp = \amp 0,\\
2x_1 \amp + \amp 4x_2 \amp - \amp 5x_3 \amp + \amp 8x_4 \amp = \amp 0.
\end{array}\right.
\end{equation*}
Careful. After you’ve reduced the homogeneous system in this activity, remember that there is still the omitted “equals” column of all zeros.
(a) There is one obvious particular solution to the system. What is it?
(b) Will any row operation ever alter the “equals” column?
(c) Convert the system to a coefficient matrix (i.e. omit the “equals” column). Then solve as usual.
Discovery 2.5 .
In a homogeneous system, what is the relationship between the number of variables, the number of “leading ones” in the most reduced form of the coefficient matrix, and the number of parameters required to solve the system? What pattern of leading ones in a completely reduced coefficient matrix tells you that the corresponding homogeneous system has a single, unique solution?
Discovery 2.6 .
Consider system
\begin{equation*}
\left\{\begin{array}{rcrcrcrcrcr}
3x_1 \amp - \amp x_2 \amp + \amp 4x_3 \amp = \amp b_1,\\
x_1 \amp + \amp 2x_2 \amp - \amp x_3 \amp = \amp b_2,\\
3x_1 \amp \amp \amp + \amp 3x_3 \amp = \amp b_3,
\end{array}\right.
\end{equation*}
where the constants of each equation are not specified. For what values of the unknown constants \(b_1,b_2,b_3\) is this system consistent?
To answer this question, row reduce the associated augmented matrix (below) until you are at a point where you can determine conditions on the constants \(b_1,b_2,b_3\) that ensures the system is consistent.
\begin{equation*}
\left[\begin{array}{rrr|r}
3 \amp -1 \amp 4 \amp b_1\\1 \amp 2 \amp -1 \amp b_2\\3 \amp 0 \amp 3 \amp b_3
\end{array}\right].
\end{equation*}
