2.1E: Elementary Matrices Exercises

Exercises for 1

solutions

2

For each of the following elementary matrices, describe the corresponding elementary row operation and write the inverse.

$E = \left[ \begin{array}{rrr} 1 & 0 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$ $E = \left[ \begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right]$ $E = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{array} \right]$ $E = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$ $E = \left[ \begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right]$ $E = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array} \right]$

2. Interchange rows 1 and 3 of $I$. $E^{-1} = E$.
3. Add $(-2)$ times row 1 of $I$ to row 2. $E^{-1} = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$
4. Multiply row 3 of $I$ by $5$. $E^{-1} = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \frac{1}{5} \end{array} \right]$

In each case find an elementary matrix $E$ such that $B = EA$.

1. $A = \left[ \begin{array}{rr} 2 & 1 \\ 3 & -1 \end{array} \right]$, $B = \left[ \begin{array}{rr} 2 & 1 \\ 1 & -2 \end{array} \right]$
2. $A = \left[ \begin{array}{rr} -1 & 2 \\ 0 & 1 \end{array} \right]$, $B = \left[ \begin{array}{rr} 1 & -2 \\ 0 & 1 \end{array} \right]$
3. $A = \left[ \begin{array}{rr} 1 & 1 \\ -1 & 2 \end{array} \right]$, $B = \left[ \begin{array}{rr} -1 & 2 \\ 1 & 1 \end{array} \right]$
4. $A = \left[ \begin{array}{rr} 4 & 1 \\ 3 & 2 \end{array} \right]$, $B = \left[ \begin{array}{rr} 1 & -1 \\ 3 & 2 \end{array} \right]$
5. $A = \left[ \begin{array}{rr} -1 & 1 \\ 1 & -1 \end{array} \right]$, $B = \left[ \begin{array}{rr} -1 & 1 \\ -1 & 1 \end{array} \right]$
6. $A = \left[ \begin{array}{rr} 2 & 1 \\ -1 & 3 \end{array} \right]$, $B = \left[ \begin{array}{rr} -1 & 3 \\ 2 & 1 \end{array} \right]$

2. $\left[ \begin{array}{rr} -1 & 0 \\ 0 & 1 \end{array} \right]$
3. $\left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array} \right]$
4. $\left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]$

Let $A = \left[ \begin{array}{rr} 1 & 2 \\ -1 & 1 \end{array} \right]$ and
$C = \left[ \begin{array}{rr} -1 & 1 \\ 2 & 1 \end{array} \right]$.

1. Find elementary matrices $E_{1}$ and $E_{2}$ such that $C = E_{2}E_{1}A$.
2. Show that there is no elementary matrix $E$ such that $C = EA$.

2. The only possibilities for $E$ are $\left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]$, $\left[ \begin{array}{rr} k & 0 \\ 0 & 1 \end{array} \right]$, $\left[ \begin{array}{rr} 1 & 0 \\ 0 & k \end{array} \right]$, $\left[ \begin{array}{rr} 1 & k \\ 0 & 1 \end{array} \right]$, and $\left[ \begin{array}{rr} 1 & 0 \\ k & 1 \end{array} \right]$. In each case, $EA$ has a row different from $C$.

If $E$ is elementary, show that $A$ and $EA$ differ in at most two rows.

1. Is $I$ an elementary matrix? Explain.
2. Is $0$ an elementary matrix? Explain.

2. No, $0$ is not invertible.

In each case find an invertible matrix $U$ such that $UA = R$ is in reduced row-echelon form, and express $U$ as a product of elementary matrices.

$A = \left[ \begin{array}{rrr} 1 & -1 & 2 \\ -2 & 1 & 0 \end{array} \right]$ $A = \left[ \begin{array}{rrr} 1 & 2 & 1 \\ 5 & 12 & -1 \end{array} \right]$ $A = \left[ \begin{array}{rrrr} 1 & 2 & -1 & 0 \\ 3 & 1 & 1 & 2 \\ 1 & -3 & 3 & 2 \end{array} \right]$ $A = \left[ \begin{array}{rrrr} 2 & 1 & -1 & 0 \\ 3 & -1 & 2 & 1 \\ 1 & -2 & 3 & 1 \end{array} \right]$

2. $\left[ \begin{array}{rr} 1 & -2 \\ 0 & 1 \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \\ 0 & \frac{1}{2} \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \\ -5 & 1 \end{array} \right]$
   $A = \left[ \begin{array}{rrr} 1 & 0 & 7 \\ 0 & 1 & -3 \end{array} \right]$. Alternatively,
   $\left[ \begin{array}{rr} 1 & 0 \\ 0 & \frac{1}{2} \end{array} \right] \left[ \begin{array}{rr} 1 & -1 \\ 0 & 1 \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \\ -5 & 1 \end{array} \right]$
   $A = \left[ \begin{array}{rrr} 1 & 0 & 7 \\ 0 & 1 & -3 \end{array} \right]$.

3. $\left[ \begin{array}{rrr} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & \frac{1}{5} & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{array} \right]$
   $\left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -2 & 0 & 1 \end{array} \right] \left[ \begin{array}{rrr} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$

   $\left[ \begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right] A = \left[ \def\arraystretch{1.5}\begin{array}{rrrr} 1 & 0 & \frac{1}{5} & \frac{1}{5} \\ 0 & 1 & -\frac{7}{5} & -\frac{2}{5} \\ 0 & 0 & 0 & 0 \end{array} \right]$

In each case find an invertible matrix $U$ such that $UA = B$, and express $U$ as a product of elementary matrices.

1. $A = \left[ \begin{array}{rrr} 2 & 1 & 3 \\ -1 & 1 & 2 \end{array} \right]$, $B = \left[ \begin{array}{rrr} 1 & -1 & -2 \\ 3 & 0 & 1 \end{array} \right]$
2. $A = \left[ \begin{array}{rrr} 2 & -1 & 0 \\ 1 & 1 & 1 \end{array} \right]$, $B = \left[ \begin{array}{rrr} 3 & 0 & 1 \\ 2 & -1 & 0 \end{array} \right]$

2. $U = \left[ \begin{array}{rr} 1 & 1 \\ 1 & 0 \end{array} \right] = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right] \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]$

In each case factor $A$ as a product of elementary matrices.

$A = \left[ \begin{array}{rr} 1 & 1 \\ 2 & 1 \end{array} \right]$ $A = \left[ \begin{array}{rr} 2 & 3 \\ 1 & 2 \end{array} \right]$ $A = \left[ \begin{array}{rrr} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 2 & 1 & 6 \end{array} \right]$ $A = \left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 4 \\ -2 & 2 & 15 \end{array} \right]$

2. $A = \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array} \right] \left[ \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right]$
   $\left[ \begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array} \right]$

3. $A = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -2 & 0 & 1 \end{array} \right] \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{array} \right]$
   $\left[ \begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{array} \right]$

[ex:ex2_5_9] Let $E$ be an elementary matrix.

1. Show that $E^{T}$ is also elementary of the same type.
2. Show that $E^{T} = E$ if $E$ is of type I or II.

Show that every matrix $A$ can be factored as $A = UR$ where $U$ is invertible and $R$ is in reduced row-echelon form.

$UA = R$ by Theorem [thm:005294], so $A = U^{-1}R$.

If $A = \left[ \begin{array}{rr} 1 & 2 \\ 1 & -3 \end{array} \right]$ and
$B = \left[ \begin{array}{rr} 5 & 2 \\ -5 & -3 \end{array} \right]$ find an elementary matrix $F$ such that $AF = B$.

[Hint: See Exercise [ex:ex2_5_9].]

In each case find invertible $U$ and $V$ such that $UAV = \left[ \begin{array}{cc} I_{r} & 0 \\ 0 & 0 \end{array} \right]$, where $r = rank \;A$.

$A = \left[ \begin{array}{rrr} 1 & 1 & -1 \\ -2 & -2 & 4 \end{array} \right]$ $A = \left[ \begin{array}{rr} 3 & 2 \\ 2 & 1 \end{array} \right]$ $A = \left[ \begin{array}{rrrr} 1 & -1 & 2 & 1 \\ 2 & -1 & 0 & 3 \\ 0 & 1 & -4 & 1 \end{array} \right]$ $A = \left[ \begin{array}{rrrr} 1 & 1 & 0 & -1 \\ 3 & 2 & 1 & 1 \\ 1 & 0 & 1 & 3 \end{array} \right]$

2. $U = A^{-1}$, $V = I^{2}$; $rank \;A = 2$

3. $U = \left[ \begin{array}{rrr} -2 & 1 & 0 \\ 3 & -1 & 0 \\ 2 & -1 & 1 \end{array} \right]$,
   $V = \left[ \begin{array}{rrrr} 1 & 0 & -1 & -3 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$; $rank \;A = 2$

Prove Lemma [lem:005213] for elementary matrices of:

type I; type II.

While trying to invert $A$, $\left[ \begin{array}{cc} A & I \end{array} \right]$ is carried to $\left[ \begin{array}{cc} P & Q \end{array} \right]$ by row operations. Show that $P = QA$.

If $A$ and $B$ are $n \times n$ matrices and $AB$ is a product of elementary matrices, show that the same is true of $A$.

If $U$ is invertible, show that the reduced row-echelon form of a matrix $\left[ \begin{array}{cc} U & A \end{array} \right]$ is $\left[ \begin{array}{cc} I & U^{-1}A \end{array} \right]$.

Write $U^{-1} = E_{k}E_{k-1} \cdots E_{2}E_{1}$, $E_{i}$ elementary. Then $\left[ \begin{array}{cc} I & U^{-1}A \end{array} \right] = \left[ \begin{array}{cc} U^{-1}U & U^{-1}A \end{array} \right]$
$= U^{-1} \left[ \begin{array}{cc} U & A \end{array} \right] = E_{k}E_{k-1} \cdots E_{2}E_{1} \left[ \begin{array}{cc} U & A \end{array} \right]$. So $\left[ \begin{array}{cc} U & A \end{array} \right] \rightarrow \left[ \begin{array}{cc} I & U^{-1}A \end{array} \right]$ by row operations
(Lemma [lem:005213]).

[ex:ex2_5_17] Two matrices $A$ and $B$ are called row-equivalent (written $A \overset{r}{\sim} B$) if there is a sequence of elementary row operations carrying $A$ to $B$.

1. Show that $A \overset{r}{\sim} B$ if and only if $A = UB$ for some invertible matrix $U$.
2. Show that:
   1. $A \overset{r}{\sim} A$ for all matrices $A$.
   2. If $A \overset{r}{\sim} B$, then $B \overset{r}{\sim} A$
   3. If $A \overset{r}{\sim} B$ and $B \overset{r}{\sim} C$, then $A \overset{r}{\sim} C$.
3. Show that, if $A$ and $B$ are both row-equivalent to some third matrix, then $A \overset{r}{\sim} B$.

4. Show that $\left[ \begin{array}{rrrr} 1 & -1 & 3 & 2 \\ 0 & 1 & 4 & 1 \\ 1 & 0 & 8 & 6 \end{array} \right]$ and
   $\left[ \begin{array}{rrrr} 1 & -1 & 4 & 5 \\ -2 & 1 & -11 & -8 \\ -1 & 2 & 2 & 2 \end{array} \right]$ are row-equivalent. [Hint: Consider (c) and Theorem [thm:001017].]

2. (i) $A \overset{r}{\sim} A$ because $A = IA$. (ii) If $A \overset{r}{\sim} B$, then $A = UB$, $U$ invertible, so $B = U^{-1}A$. Thus $B \overset{r}{\sim} A$. (iii) If $A \overset{r}{\sim} B$ and $B \overset{r}{\sim} C$, then $A = UB$ and $B = VC$, $U$ and $V$ invertible. Hence $A = U(VC) = (UV)C$, so $A \overset{r}{\sim} C$.

If $U$ and $V$ are invertible $n \times n$ matrices, show that $U \overset{r}{\sim} V$. (See Exercise [ex:ex2_5_17].)

(See Exercise [ex:ex2_5_17].) Find all matrices that are row-equivalent to:

$\left[ \begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]$ $\left[ \begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right]$ $\left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right]$ $\left[ \begin{array}{rrr} 1 & 2 & 0 \\ 0 & 0 & 1 \end{array} \right]$

2. If $B \overset{r}{\sim} A$, let $B = UA$, $U$ invertible. If $U = \left[ \begin{array}{rr} d & b \\ -b & d \end{array} \right]$, $B = UA = \left[ \begin{array}{ccc} 0 & 0 & b \\ 0 & 0 & d \end{array} \right]$ where $b$ and $d$ are not both zero (as $U$ is invertible). Every such matrix $B$ arises in this way: Use $U = \left[ \begin{array}{rr} a & b \\ -b & a \end{array} \right]$–it is invertible by Example [exa:003540].

Let $A$ and $B$ be $m \times n$ and $n \times m$ matrices, respectively. If $m > n$, show that $AB$ is not invertible. [Hint: Use Theorem [thm:001473] to find $\mathbf{x} \neq \mathbf{0}$ with $B\mathbf{x} = \mathbf{0}$.]

Define an elementary column operation on a matrix to be one of the following: (I) Interchange two columns. (II) Multiply a column by a nonzero scalar. (III) Add a multiple of a column to another column. Show that:

1. If an elementary column operation is done to an $m \times n$ matrix $A$, the result is $AF$, where $F$ is an $n \times n$ elementary matrix.
2. Given any $m \times n$ matrix $A$, there exist $m \times m$ elementary matrices $E_{1}, \dots, E_{k}$ and $n \times n$ elementary matrices $F_{1}, \dots, F_{p}$ such that, in block form,

   $$E_{k} \cdots E_{1}AF_{1} \cdots F_{p} = \left[ \begin{array}{cc} I_{r} & 0 \\ 0 & 0 \end{array} \right] \nonumber$$

Suppose $B$ is obtained from $A$ by:

1. interchanging rows $i$ and $j$;
2. multiplying row $i$ by $k \neq 0$;
3. adding $k$ times row $i$ to row $j\ (i \neq j)$.

In each case describe how to obtain $B^{-1}$ from $A^{-1}$. [Hint: See part (a) of the preceding exercise.]

2. Multiply column $i$ by $1/k$.

Two $m \times n$ matrices $A$ and $B$ are called equivalent (written $A \overset{e}{\sim} B$) if there exist invertible matrices $U$ and $V$ (sizes $m \times m$ and $n \times n$) such that $A = UBV$.

1. Prove the following the properties of equivalence.
   1. $A \overset{e}{\sim} A$ for all $m \times n$ matrices $A$.
   2. If $A \overset{e}{\sim} B$, then $B \overset{e}{\sim} A$.
   3. If $A \overset{e}{\sim} B$ and $B \overset{e}{\sim} C$, then $A \overset{e}{\sim} C$.
2. Prove that two $m \times n$ matrices are equivalent if they have the same $rank \;$. [Hint: Use part (a) and Theorem [thm:005369].]