Exercise RREF.C05 Robert Beezer

Each archetype below is a system of equations. Form the augmented matrix of the system of equations, convert the matrix to reduced row-echelon form by using equation operations and then describe the solution set of the original system of equations.

Archetype A, Archetype B, Archetype C, Archetype D, Archetype E, Archetype F, Archetype G, Archetype H, Archetype I, Archetype J

For problems C10–C19, find all solutions to the system of linear equations. Use your favorite computing device to row-reduce the augmented matrices for the systems, and write the solutions as a set, using correct set notation.

Exercise RREF.C10 Robert Beezer

\begin{align*} 2x_1-3x_2+x_3+7x_4&=14\\ 2x_1+8x_2-4x_3+5x_4&=-1\\ x_1+3x_2-3x_3&=4 \\ -5x_1+2x_2+3x_3+4x_4&=-19 \end{align*}

Solution

Exercise RREF.C11 Robert Beezer

\begin{align*} 3x_1+4x_2-x_3+2x_4&=6\\ x_1-2x_2+3x_3+x_4&=2\\ 10x_2-10x_3-x_4&=1 \end{align*}

Solution

Exercise RREF.C12 Robert Beezer

\begin{align*} 2x_1+4x_2+5x_3+7x_4&=-26\\ x_1+2x_2+x_3-x_4&=-4\\ -2x_1-4x_2+x_3+11x_4&=-10 \end{align*}

Solution

Exercise RREF.C13 Robert Beezer

\begin{align*} x_1+2x_2+8x_3-7x_4&=-2\\ 3x_1+2x_2+12x_3-5x_4&=6\\ -x_1+x_2+x_3-5x_4&=-10 \end{align*}

Solution

Exercise RREF.C14 Robert Beezer

\begin{align*} 2x_1+x_2+7x_3-2x_4&=4\\ 3x_1-2x_2+11x_4&=13\\ x_1+x_2+5x_3-3x_4&=1 \end{align*}

Solution

Exercise RREF.C15 Robert Beezer

\begin{align*} 2x_1+ 3x_2-x_3 -9x_4&=-16\\ x_1+ 2x_2+ x_3&=0\\ -x_1+ 2x_2+ 3x_3+ 4x_4&=8 \end{align*}

Solution

Exercise RREF.C16 Robert Beezer

\begin{align*} 2x_1+ 3x_2+19x_3 -4x_4&=2\\ x_1+ 2x_2+ 12x_3-3x_4&=1\\ -x_1+ 2x_2+ 8x_3-5x_4&=1 \end{align*}

Solution

Exercise RREF.C17 Robert Beezer

\begin{align*} -x_1+5x_2&=-8\\ -2x_1+5x_2+5x_3+2x_4&=9\\ -3x_1-x_2+3x_3+x_4&=3\\ 7x_1+6x_2+5x_3+x_4&=30 \end{align*}

Solution

Exercise RREF.C18 Robert Beezer

\begin{align*} x_1+2x_2-4x_3-x_4&=32\\ x_1+3x_2-7x_3-x_5&=33\\ x_1+2x_3-2x_4+3x_5&=22\\ \end{align*}

Solution

Exercise RREF.C19 Robert Beezer

\begin{align*} 2x_1 + x_2 &= 6 \\ -x_1 - x_2 &= -2 \\ 3x_1 + 4x_2 &= 4 \\ 3x_1 + 5x_2 &= 2 \end{align*}

Solution

For problems C30–C33, row-reduce the matrix without the aid of a calculator, indicating the row operations you are using at each step using the notation of Definition RO.

Exercise RREF.C30 Robert Beezer

\begin{align*} \begin{bmatrix} 2 & 1 & 5 & 10\\ 1 & -3 & -1 & -2\\ 4 & -2 & 6 & 12 \end{bmatrix} \end{align*}

Solution

Exercise RREF.C31 Robert Beezer

\begin{align*} \begin{bmatrix} 1 & 2 & -4 \\ -3 & -1 & -3 \\ -2 & 1 & -7 \end{bmatrix} \end{align*}

Solution

Exercise RREF.C32 Robert Beezer

\begin{align*} \begin{bmatrix} 1 & 1 & 1 \\ -4 & -3 & -2 \\ 3 & 2 & 1 \end{bmatrix} \end{align*}

Solution

Exercise RREF.C33 Robert Beezer

\begin{align*} \begin{bmatrix} 1 & 2 & -1 & -1 \\ 2 & 4 & -1 & 4 \\ -1 & -2 & 3 & 5 \end{bmatrix} \end{align*}

Solution

Exercise RREF.M40 Robert Beezer

Consider the two $3\times 4$ matrices below \begin{align*} B&= \begin{bmatrix} 1 & 3 & -2 & 2 \\ -1 & -2 & -1 & -1 \\ -1 & -5 & 8 & -3 \end{bmatrix} & C&= \begin{bmatrix} 1 & 2 & 1 & 2 \\ 1 & 1 & 4 & 0 \\ -1 & -1 & -4 & 1 \end{bmatrix} \end{align*}

  1. Row-reduce each matrix and determine that the reduced row-echelon forms of $B$ and $C$ are identical. From this argue that $B$ and $C$ are row-equivalent.
  2. In the proof of Theorem RREFU, we begin by arguing that entries of row-equivalent matrices are related by way of certain scalars and sums. In this example, we would write that entries of $B$ from row $i$ that are in column $j$ are linearly related to the entries of $C$ in column $j$ from all three rows \begin{align*} \matrixentry{B}{ij} &= \delta_{i1}\matrixentry{C}{1j}+ \delta_{i2}\matrixentry{C}{2j}+ \delta_{i3}\matrixentry{C}{3j} & 1&\leq j\leq 4 \end{align*} For each $1\leq i\leq 3$ find the corresponding three scalars in this relationship. So your answer will be nine scalars, determined three at a time.

Solution

Exercise RREF.M45 Chris Black

You keep a number of lizards, mice and peacocks as pets. There are a total of 108 legs and 30 tails in your menagerie. You have twice as many mice as lizards. How many of each creature do you have?

Solution

Exercise RREF.M50 Robert Beezer

A parking lot has 66 vehicles (cars, trucks, motorcycles and bicycles) in it. There are four times as many cars as trucks. The total number of tires (4 per car or truck, 2 per motorcycle or bicycle) is 252. How many cars are there? How many bicycles?

Solution

Exercise RREF.T10 Robert Beezer

Prove that each of the three row operations (Definition RO) is reversible. More precisely, if the matrix $B$ is obtained from $A$ by application of a single row operation, show that there is a single row operation that will transform $B$ back into $A$.

Solution

Exercise RREF.T11 Robert Beezer

Suppose that $A$, $B$ and $C$ are $m\times n$ matrices. Use the definition of row-equivalence (Definition REM) to prove the following three facts.

  1. $A$ is row-equivalent to $A$.
  2. If $A$ is row-equivalent to $B$, then $B$ is row-equivalent to $A$.
  3. If $A$ is row-equivalent to $B$, and $B$ is row-equivalent to $C$, then $A$ is row-equivalent to $C$.
A relationship that satisfies these three properties is known as an equivalence relation, an important idea in the study of various algebras. This is a formal way of saying that a relationship behaves like equality, without requiring the relationship to be as strict as equality itself. We will see it again in Theorem SER.

Exercise RREF.T12 Robert Beezer

Suppose that $B$ is an $m\times n$ matrix in reduced row-echelon form. Build a new, likely smaller, $k\times\ell$ matrix $C$ as follows. Keep any collection of $k$ adjacent rows, $k\leq m$. From these rows, keep columns $1$ through $\ell$, $\ell\leq n$. Prove that $C$ is in reduced row-echelon form.

Exercise RREF.T13 Robert Beezer

Generalize Exercise RREF.T12 by just keeping any $k$ rows, and not requiring the rows to be adjacent. Prove that any such matrix $C$ is in reduced row-echelon form.