# 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. 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 $d-cb \neq 0$.

2. 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)

3. 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
$\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).$