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
    \[\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).\]