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].]