Objectives
- Solve linear systems using row operations on augmented matrices
- Recognize and use the elementary row operations and their notations
Worksheet Participate
1. Row Operations Notation.
For the matrix
\begin{equation*}
A=\left[\begin{array}{ccc} 2\amp -1\amp 1\\ 4\amp 0\amp 5\\ -1\amp 1\amp 1\\ \end{array} \right]\text{,}
\end{equation*}
give the matrix that results from performing each of the row operations below.
(a)
\(-2R_1+R_2 \rightarrow R_2\text{.}\)
(b)
\(\frac{1}{2}R_2 \rightarrow R_2\text{.}\)
(c)
\(R_1 \leftrightarrow R_3\text{.}\)
2. Side-by-side Elimination and Matrices.
Solve the linear system below using the method of elimination side-by-side with row operations.
\begin{align*}
2x_1 + 4x_2 \amp = -6\\
5x_1 - 5x_2 \amp = 15
\end{align*}
\begin{equation*}
\begin{bmatrix} 2 \amp 4 \amp -6 \\ 5 \amp -5 \amp 15 \\ \end{bmatrix}
\end{equation*}
What row operation was done?
Write down the row operation that transforms \(A\) into the matrix given in each exercise.
\begin{equation*}
A=\left[\begin{array}{ccc} 2\amp -1\amp 1\\ 4\amp 0\amp 5\\ -1\amp 1\amp 1\\ \end{array} \right]
\end{equation*}
3.
\begin{equation*}
\left[\begin{array}{ccc} 2\amp -1\amp 1\\ 4\amp 0\amp 5\\ 2\amp -2\amp -2\\ \end{array} \right]
\end{equation*}
4.
\begin{equation*}
\left[\begin{array}{ccc} 0\amp 1\amp 3\\ 4\amp 0\amp 5\\ -1\amp 1\amp 1\\ \end{array} \right]
\end{equation*}
5.
\begin{equation*}
\left[\begin{array}{ccc} 2\amp -1\amp 1\\ -1\amp 1\amp 1\\ 4\amp 0\amp 5\\ \end{array} \right]
\end{equation*}
6. Two truths and a lie?
Using shorthand row operation notation, write down two elementary row operations and one thing that looks like but isn’t an elementary row operation.
7. Exploring non-elementary row operations.
(a)
Do something to the matrix in Problem 2 that is not an elementary row operation, and write down the matrix that you get.
(b)
Solve the linear system corresponding to your new matrix. Does the linear system corresponding to your new matrix have the same solution as the linear system in Problem 2?
(c)
If the linear system has the same solution, can you write down a sequence of elementary row operations that results in your new matrix?
