Describing the Solution Set

Find the indicated entry of the matrix, if it is defined.

• (a)  $a_{2,1}$     (b)  $a_{1,2}$     (c)  $a_{2,2}$     (d)  $a_{3,1}$

Give the size of each matrix.

• (a)  $\begin{pmatrix}1&0&4\\ 2&1&5\end{pmatrix}$     (b)  $\begin{pmatrix}1&1\\ -1&1\\ 3&-1\end{pmatrix}$     (c)  $\begin{pmatrix}5&10\\ 10&5\end{pmatrix}$

Do the indicated vector operation, if it is defined.

• (a)  $\begin{pmatrix}2\\ 1\\ 1\end{pmatrix}+\begin{pmatrix}3\\ 0\\ 4\end{pmatrix}$     (b)  $5\begin{pmatrix}4\\ -1\end{pmatrix}$     (c)  $\begin{pmatrix}1\\ 5\\ 1\end{pmatrix}-\begin{pmatrix}3\\ 1\\ 1\end{pmatrix}$     (d)  $7\begin{pmatrix}2\\ 1\end{pmatrix}+9\begin{pmatrix}3\\ 5\end{pmatrix}$     (e)  $\begin{pmatrix}1\\ 2\end{pmatrix}+\begin{pmatrix}1\\ 2\\ 3\end{pmatrix}$     (f)  $6\begin{pmatrix}3\\ 1\\ 1\end{pmatrix}-4\begin{pmatrix}2\\ 0\\ 3\end{pmatrix}+2\begin{pmatrix}1\\ 1\\ 5\end{pmatrix}$

Solve each system using matrix notation. Express the solution using vectors.

• (a)  $\begin{array}{rcrcr}3x&+&6y&=&18\\ x&+&2y&=&6\end{array}$     (b)  $\begin{array}{rcrcr}x&+&y&=&1\\ x&-&y&=&-1\end{array}$     (c)  $\begin{array}{rcrcrcr}x_{1}&&&+&x_{3}&=&4\\ x_{1}&-&x_{2}&+&2x_{3}&=&5\\ 4x_{1}&-&x_{2}&+&5x_{3}&=&17\end{array}$     (d)  $\begin{array}{rcrcrcr}2a&+&b&-&c&=&2\\ 2a&&&+&c&=&3\\ a&-&b&&&=&0\end{array}$     (e)  $\begin{array}{rcrcrcrcr}x&+&2y&-&z&&&=&3\\ 2x&+&y&&&+&w&=&4\\ x&-&y&+&z&+&w&=&1\end{array}$     (f)  $\begin{array}{rcrcrcrcr}x&&&+&z&+&w&=&4\\ 2x&+&y&&&-&w&=&2\\ 3x&+&y&+&z&&&=&7\end{array}$

Solve each system using matrix notation. Give each solution set in vector notation.

• (a)  $\begin{array}{rcrcrcr}2x&+&y&-&z&=&1\\ 4x&-&y&&&=&3\end{array}$     (b)  $\begin{array}{rcrcrcrcr}x&&&-&z&&&=&1\\ &&y&+&2z&-&w&=&3\\ x&+&2y&+&3z&-&w&=&7\end{array}$     (c)  $\begin{array}{rcrcrcrcr}x&-&y&+&z&&&=&0\\ &&y&&&+&w&=&0\\ 3x&-&2y&+&3z&+&w&=&0\\ &&-y&&&-&w&=&0\end{array}$     (d)  $\begin{array}{rcrcrcrcrcr}a&+&2b&+&3c&+&d&-&e&=&1\\ 3a&-&b&+&c&+&d&+&e&=&3\end{array}$

Solve each system using matrix notation. Express the solution set using vectors.

• (a)  $\begin{array}{rcrcrcr}3x&+&2y&+&z&=&1\\ x&-&y&+&z&=&2\\ 5x&+&5y&+&z&=&0\end{array}$     (b)  $\begin{array}{rcrcrcrcr}x&+&y&-&2z&=&0\\ x&-&y&&&=&-3\\ 3x&-&y&-&2z&=&-6\\ &&2y&-&2z&=&3\end{array}$     (c)  $\begin{array}{rcrcrcrcrcr}2x&-&y&-&z&+&w&=&4\\ x&+&y&+&z&&&=&-1\end{array}$     (d)  $\begin{array}{rcrcrcr}x&+&y&-&2z&=&0\\ x&-&y&&&=&-3\\ 3x&-&y&-&2z&=&0\end{array}$

The vector is in the set. What value of the parameters produces that vector?

• $\begin{pmatrix}5\\ -5\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}1\\ -1\end{pmatrix}k\mid k\in\mathbb{R}\hskip 1.00006pt\}$

• $\begin{pmatrix}-1\\ 2\\ 1\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}-2\\ 1\\ 0\end{pmatrix}i+\begin{pmatrix}3\\ 0\\ 1\end{pmatrix}j\mid i,j\in\mathbb{R}\hskip 1.00006pt\}$

• $\begin{pmatrix}0\\ -4\\ 2\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}m+\begin{pmatrix}2\\ 0\\ 1\end{pmatrix}n\mid m,n\in\mathbb{R}\hskip 1.00006pt\}$

Decide if the vector is in the set.

• $\begin{pmatrix}3\\ -1\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}-6\\ 2\end{pmatrix}k\mid k\in\mathbb{R}\hskip 1.00006pt\}$

• $\begin{pmatrix}5\\ 4\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}5\\ -4\end{pmatrix}j\mid j\in\mathbb{R}\hskip 1.00006pt\}$

• $\begin{pmatrix}2\\ 1\\ -1\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}0\\ 3\\ -7\end{pmatrix}+\begin{pmatrix}1\\ -1\\ 3\end{pmatrix}r\mid r\in\mathbb{R}\hskip 1.00006pt\}$

• $\begin{pmatrix}1\\ 0\\ 1\end{pmatrix}$, $\{\hskip 1.00006pt\begin{pmatrix}2\\ 0\\ 1\end{pmatrix}j+\begin{pmatrix}-3\\ -1\\ 1\end{pmatrix}k\mid j,k\in\mathbb{R}\hskip 1.00006pt\}$

A farmer with 1200 acres is considering planting three different crops, corn, soybeans, and oats. The farmer wants to use all $1200$ acres. Seed corn costs $$20$ per acre, while soybean and oat seed cost $$50$ and $$12$ per acre respectively. The farmer has $$40\,000$ available to buy seed and intends to spend it all.

• Use the information above to formulate two linear equations with three unknowns and solve it.

• Sol…

Parametrize the solution set of this one-equation system.

• Apply Gauss’s Method to the left-hand side to solve

  for $x$, $y$, $z$, and $w$, in terms of the constants $a$, $b$, and $c$.

• Use your answer from the prior part to solve this.

Why is the comma needed in the notation ‘$a_{i,j}$’ for matrix entries?

Give the $4\!\times\!4$ matrix whose $i,j$-th entry is

• (a)  $i+j$;     (b)  $-1$ to the $i+j$ power.

For any matrix $A$, the transpose of $A$, written ${{A}^{\mathsf{T}}}$, is the matrix whose columns are the rows of $A$. Find the transpose of each of these.

• (a)  $\begin{pmatrix}1&2&3\\ 4&5&6\end{pmatrix}$     (b)  $\begin{pmatrix}2&-3\\ 1&1\end{pmatrix}$     (c)  $\begin{pmatrix}5&10\\ 10&5\end{pmatrix}$     (d)  $\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}$

• Describe all functions $f(x)=ax^{2}+bx+c$ such that $f(1)=2$ and $f(-1)=6$.

• Describe all functions $f(x)=ax^{2}+bx+c$ such that $f(1)=2$.

Show that any set of five points from the plane $\mathbb{R}^{2}$ lie on a common conic section, that is, they all satisfy some equation of the form $ax^{2}+by^{2}+cxy+dx+ey+f=0$ where some of $a,\,\ldots\,,f$ are nonzero.

Make up a four equations/four unknowns system having

• a one-parameter solution set;

• a two-parameter solution set;

• a three-parameter solution set.

This puzzle is from a Russian web-site http://www.arbuz.uz/ and there are many solutions to it, but mine uses linear algebra and is very naive. There’s a planet inhabited by arbuzoids (watermeloners, to translate from Russian). Those creatures are found in three colors: red, green and blue. There are $13$ red arbuzoids, $15$ blue ones, and $17$ green. When two differently colored arbuzoids meet, they both…

• Solve the system of equations.

  For what values of $a$ does the system fail to have solutions, and for what values of $a$ are there infinitely many solutions?

• Answer the above question for the system.

In air a gold-surfaced sphere weighs $7588$ grams. It is known that it may contain one or more of the metals aluminum, copper, silver, or lead. When weighed successively under standard conditions in water, benzene, alcohol, and glycerin its respective weights are $6588$, $6688$, $6778$, and $6328$ grams. How much, if any, of the forenamed metals does it contain if the specific gravities of the designated substances are t…