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Section 3.9 Row Operations via Matrix Multiplication: Homework
Exercises Exercises
1.
In a system of equations represented by a matrix \(A \in \R^{3 \times 4}\) write down the elementary matrix that corresponds to each row operation.
\(R_1\) swapped with \(R_3\text{.}\)
\(-3R_2+5R_3 \rightarrow R_3\text{.}\)
The row operation in part (a) followed by the row operation in part (b).
The row operation in part (b) followed by the row operation in part (a).
Did you get the same answer for (c) and (d). Are you surprised? Why or why not?
2.
Consider the following system:
\begin{equation*}
\begin{array}{rrcrcrcr}
R_1:\, \amp -x \amp + \amp 2y \amp + \amp 4z \amp = \amp 1\\
R_2:\, \amp 3x\amp +\amp y \amp - \amp 2z\amp =\amp 13\\
R_3:\, \amp 6x\amp +\amp 2y \amp +\amp 3z\amp = \amp 47\amp
\end{array}
\end{equation*}
Write down the augmented matrix \(A\) that corresponds to the system.
Perform elementary row operations to reduce the system to echelon form. For each step keep track of the corresponding elementary matrix \(E_1,...,E_m\text{.}\)
Multiply the elementary matrices together in reverse order to get \(E=E_m\cdots E_2E_1\text{.}\)
Perform the multiplication \(EA\) and verify it corresponds to the row-echelon form you found in part (b).
3.
For each of the following elementary matrices in \(R^{3 \times 3}\text{,}\) find the inverse using a guess and check method. Show the check that your answer is correct (that is, \(EE^{-1}=I\) ).
\(\displaystyle E_{23}\)
\(\displaystyle E_{(2)11}\)
\(\displaystyle E_{(-2)32}\)
4.
For each of the following elementary matrices in \(R^{4 \times 4}\text{,}\) find the inverse using a guess and check method. Show the check that your answer is correct (that is, \(EE^{-1}=I\) ).
\(\displaystyle E_{24}\)
\(\displaystyle E_{(-1/2)33}\)
\(\displaystyle E_{(4)34}\)
