Gauss-Jordan Reduction

Use Gauss-Jordan reduction to solve each system.

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

Do Gauss-Jordan reduction.

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

Find the reduced echelon form of each matrix.

• (a)  $\begin{pmatrix}2&1\\ 1&3\end{pmatrix}$     (b)  $\begin{pmatrix}1&3&1\\ 2&0&4\\ -1&-3&-3\end{pmatrix}$     (c)  $\begin{pmatrix}1&0&3&1&2\\ 1&4&2&1&5\\ 3&4&8&1&2\end{pmatrix}$     (d)  $\begin{pmatrix}0&1&3&2\\ 0&0&5&6\\ 1&5&1&5\end{pmatrix}$

Get the reduced echelon form of each.

• (a)  $\begin{pmatrix}0&2&1\\ 2&-1&1\\ -2&-1&0\end{pmatrix}$     (b)  $\begin{pmatrix}1&3&1\\ 2&6&2\\ -1&0&0\end{pmatrix}$

Find each solution set by using Gauss-Jordan reduction and then reading off the parametrization.

• $\begin{array}{rcrcrcr}2x&+&y&-&z&=&1\\ 4x&-&y&&&=&3\end{array}$

• $\begin{array}{rcrcrcrcr}x&&&-&z&&&=&1\\ &&y&+&2z&-&w&=&3\\ x&+&2y&+&3z&-&w&=&7\end{array}$

• $\begin{array}{rcrcrcrcr}x&-&y&+&z&&&=&0\\ &&y&&&+&w&=&0\\ 3x&-&2y&+&3z&+&w&=&0\\ &&-y&&&-&w&=&0\end{array}$

• $\begin{array}{rcrcrcrcrcr}a&+&2b&+&3c&+&d&-&e&=&1\\ 3a&-&b&+&c&+&d&+&e&=&3\end{array}$

Give two distinct echelon form versions of this matrix.

List the reduced echelon forms possible for each size.

• (a)  $2\!\times\!2$     (b)  $2\!\times\!3$     (c)  $3\!\times\!2$     (d)  $3\!\times\!3$

What results from applying Gauss-Jordan reduction to a nonsingular matrix?

Decide whether each relation is an equivalence on the set of $2\!\times\!2$ matrices.

• two matrices are related if they have the same entry in the first row and first column

• two matrices are related if they have the same entry in the first row and first column, or the same entry in the second row and second column

Consider the following relationship on the set of $2\!\times\!2$ matrices: we say that $A$ is sum-what like $B$ if the sum of all of the entries in $A$ is the same as the sum of all the entries in $B$. For instance, the zero matrix would be sum-what like the matrix whose first row had two sevens, and whose second row had two negative sevens. Prove or disprove that this is an equivalence relation on the set of $2\!\times\!2$ matri…

The proof of Lemma 1.5 contains a reference to the $i\neq j$ condition on the row combination operation.

• Write down a $2\!\times\!2$ matrix with nonzero entries, and show that the $-1\cdot\rho_{1}+\rho_{1}$ operation is not reversed by $1\cdot\rho_{1}+\rho_{1}$.

• Expand the proof of that lemma to make explicit exactly where it uses the $i\neq j$ condition on combining.

Consider the set of students in a class. Which of the following relationships are equivalence relations? Explain each answer in at least a sentence.

• Two students $x,y$ are related if $x$ has taken at least as many math classes as $y$.

• Students $x,y$ are related if they have names that start with the same letter.

Show that each of these is an equivalence on the set of $2\!\times\!2$ matrices. Describe the equivalence classes.

• Two matrices are related if they have the same product down the diagonal, that is, if the product of the entries in the upper left and lower right are equal.

• Two matrices are related if they both have at least one entry that is a $1$, or if neither does.

Show that each is not an equivalence on the set of $2\!\times\!2$ matrices.

• Two matrices $A,B$ are related if $a_{1,1}=-b_{1,1}$.

• Two matrices are related if the sum of their entries are within $5$, that is, $A$ is related to $B$ if $\lvert(a_{1,1}+\cdots+a_{2,2})-(b_{1,1}+\cdots+b_{2,2})\rvert<5$.