# Elementary Row Operations

## Elementary Row Operations

Definition

Let $A$ be an $m\times n$ matrix.

1. The following three operations on rows of a matrix are called elementary row operations.
1.  Interchanging two rows:
$R_i \leftrightarrow R_j$ interchanges rows $i$ and $j$.
2.  Multiplying a row by a non-zero scalar:
$tR_i$ multiplies row $i$ by the non-zero scalar (number) $t$.
3. Adding a multiple of one row to another row:
$R_j+tR_i$ adds $t$ times row $i$ to row $j$.
2. Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations.
3. The matrix in reduced row echelon form that is row equivalent to $A$ is denoted by $\rref(A)$.
4. The rank of a matrix $A$ is the number of rows in $\rref(A)$.

=solution

### Problems

1. For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.
(a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$.
(b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.
(c) $C = \begin{bmatrix} 2 & -2 & 4 \\ 4 & 1 & -2 \\ 6 & -1 & 2 \end{bmatrix}$.
(d) $D = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}$.
(e) $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.

2. Let $A$ and $I$ be $2\times 2$ matrices defined as follows.
$A=\begin{bmatrix} 1 & b\\ c& d \end{bmatrix}, \qquad I=\begin{bmatrix} 1 & 0\\ 0& 1 \end{bmatrix}.$ Prove that the matrix $A$ is row equivalent to the matrix $I$ if $d-cb \neq 0$.

3. If $A, B$ have the same rank, can we conclude that they are row-equivalent? If so, then prove it. If not, then provide a counterexample.
4. Find the rank of the following real matrix.
$\begin{bmatrix} a & 1 & 2 \\ 1 &1 &1 \\ -1 & 1 & 1-a \end{bmatrix},$ where $a$ is a real number.
(Kyoto University)

5. For an $m\times n$ matrix $A$, we denote by $\mathrm{rref}(A)$ the matrix in reduced row echelon form that is row equivalent to $A$. For example, consider the matrix $A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &2 &2 \end{bmatrix}$
Then we have
$A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &2 &2 \end{bmatrix} \xrightarrow{\frac{1}{2}R_2} \begin{bmatrix} 1 & 1 & 1 \\ 0 &1 & 1 \end{bmatrix} \xrightarrow{R_1-R_2} \begin{bmatrix} 1 & 0 & 0 \\ 0 &1 &1 \end{bmatrix}$ and the last matrix is in reduced row echelon form.
Hence $\mathrm{rref}(A)=\begin{bmatrix} 1 & 0 & 0 \\ 0 &1 &1 \end{bmatrix}$. Find an example of matrices $A$ and $B$ such that
$\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).$
6. (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
(b) Find all such matrices with rank 2.

7. If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$? If so, then prove it. If not, then provide a counterexample.
8. Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.
9. Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$. Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it. Otherwise, provide a counterexample.