Objectives
-
Use the reduced row echelon form of an augmented matrix to analyze the consistency and existence of solution(s) to a linear system.
-
Identify and use properties of free and basic variables to describe solutions to linear systems.
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1.
Consider a linear system whose augmented matrix in reduced row echelon form is
\begin{equation*}
\begin{bmatrix} 1\amp 0\amp 0\amp 2\amp 3\\0\amp 0\amp 1\amp 4\amp 5\\ \end{bmatrix}\text{.}
\end{equation*}
(a)
Is the system consistent, inconsistent, or is there not enough information to tell?
(b)
Which variables are free and which are basic?
(c)
Solve the system. If there are infinitely many solutions, give two particular solutions.
2.
Consider a linear system whose augmented matrix in reduced row echelon form is
\begin{equation*}
\begin{bmatrix} 1\amp 0\amp 1\amp 4\\0\amp 1\amp 0\amp 1\\ \end{bmatrix}\text{.}
\end{equation*}
(a)
Is the system consistent, inconsistent, or is there not enough information to tell?
(b)
Which variables are free and which are basic?
(c)
Solve the system. If there are infinitely many solutions, give two particular solutions.
3.
Construct two different inconsistent linear systems with 3 variables. Use Sage or another tool to calculate the reduced row echelon form of the augmented matrix of your linear systems.
4.
Construct a linear system with 5 variables that has infinitely many solutions. Use Sage or another tool to calculate the reduced row echelon form of the augmented matrix of your linear system.
Analyzing the Effect of Different Coefficient and Constant Values.
For what values of \(k\) will the given system have exactly one solution, infinite solutions, or no solution?
5.
\begin{align*}
x_1 + 2x_2\amp =3\\
3x_1 + kx_2\amp = 9
\end{align*}
6.
\begin{align*}
x_1 + 2x_2\amp =3\\
3x_1 + kx_2\amp = 10
\end{align*}
7.
\begin{align*}
x_1 + 2x_2\amp =3\\
3x_1 + 6x_2\amp = k
\end{align*}
8.
\begin{align*}
x_1 + 2x_2\amp =3\\
3x_1 + 7x_2\amp = k
\end{align*}
