Row-reduced echelon matrices

1. Consider the following systems of equation

   1. $$\begin{array}{ccccccc} x_1 & - & x_2 & + &2x_3 & = & 1 \\ 2x_1& & & + &2x_3 & = & 1 \\ x_1 & - & 3x_2& + &4x_3 & = & 2 \end{array}$$

   2. $$\begin{array}{ccccccccc} x_1 & - & 2x_2 & + &x_3 & + & 2x_4 & = &... ...x_3 & + & x_4 & = & 2 \\ x_1 & + & 7x_2& - &5x_3 & - & x_4 & = & 3 \end{array}$$

   Find out whether they have solutions. If so, describe explicitly all solutions.
2. Let
   $${\bf A} = \left[ \begin{array}{rrr} 3 & -1 & 2 \\ 2 & 1 & 1 \\ 1 & -3 & 0 \end{array}\right].$$
   for which triples $(y_1,y_2,y_3)$ does the system ${\bf AX} = {\bf Y}$ have a solution?
3. Let
   $${\bf A} = \left[ \begin{array}{rrrr} 3 & -6 & 2 & -1 \\ -2 & 4 & 1 & 3 \\ 0 & 0 & 1 & 1 \\ 1 & -2 & 1 & 0 \\ \end{array}\right]$$
   for which $(y_1,y_2,y_3,y_4)$ does the system ${\bf AX} = {\bf Y}$ have a solution?
4. Suppose ${\bf R}$ and ${\bf R'}$ are $2 \times 3$ row-reduced echelon matrices and that the systems ${\bf RX} = {\bf0}$ and ${\bf R'X} = {\bf0}$ have exactly the same solutions. Prove that ${\bf R} = {\bf R'}$
5. Let
   $${\bf A} = \left[ \begin{array}{rrrr} 1 & 2 & 1 & 0 \\ -1 & 0 & 3 & 5 \\ 1 & -2 & 1 & 1 \\ \end{array}\right]$$
   Find a row-reduced echelon matrix ${\bf R}$ which is row-equivalent to ${\bf A}$ and an invertible $3 \times 3$ matrix ${\bf P}$ such that ${\bf R} = {\bf P}{\bf A}$.
6. For each of the three matrices
   $$\left[ \begin{array}{rrr} 2 & 5 & -1 \\ 4 & -1 & 2 \\ 6 & 4 & 1... ...4 \\ 0 & 2 & 3 & 4 \\ 0 & 0 & 3 & 4 \\ 0 & 0 & 0 & 4 \\ \end{array}\right]$$
   use elementary row operations to discover whether it is invertible, and to find the inverse in case it is.
7. Suppose ${\bf A}$ is a $2 \times 1$ matrix and that ${\bf B}$ is a $1 \times 2$ matrix. Prove that ${\bf C} = {\bf AB}$ is not invertible.
8. Let ${\bf A}$ be an $n \times n$ matrix. Prove the following:
   1. If ${\bf A}$ is invertible and ${\bf AB} = {\bf0}$ for some $n \times n$ matrix ${\bf B}$, the ${\bf B} = {\bf0}$.
   2. If ${\bf A}$ is not invertible, then there exists an $n \times n$ matrix ${\bf B}$ such that ${\bf AB} = {\bf0}$ but ${\bf B} \neq {\bf0}$.
9. Let
   $${\bf A} = \left[ \begin{array}{rr} a & b \\ c & d \\ \end{array}\right]$$
   Prove, using elementary row operations, that ${\bf A}$ is invertible if and only if $(ad - bc) \neq 0$.
10. Let
    $${\bf A} = \left[ \begin{array}{rrr} 5 & 0 & 0 \\ 1 & 5 & 0 \\ 0 & 1 & 5 \\ \end{array}\right]$$
    For which ${\bf X}$ does there exist a scalar $c$ such that ${\bf AX} = c{\bf X}$?
11. An $n \times n$ matrix ${\bf A}$ is upper-triangular if $A_{ij} = 0$ for $i > j$. Prove that ${\bf A}$ is invertible if and only if every entry on its main diagonal is distinct from 0.
12. Prove that if ${\bf A}$ is an $m \times n$ matrix, ${\bf B}$ is an $n \times m$ matrix and $n < m$, then ${\bf AB}$ is not invertible (generalization of a previous problem).
13. Let ${\bf A}$ be an $m \times n$ matrix. Show that by means of a finite number of elementary row and/or column operations one can pass from ${\bf A}$ to a matrix ${\bf R}$ which is both row-reduced echelon and column-reduced echelon, i.e., $R_{ij} = 0$ if $i \neq j$, $R_{ii} = 1, 1 \leq i \leq r$, $R_{ii} = 0, i > r$. Show that ${\bf R} = {\bf PAQ}$ where ${\bf P}$ is an invertible $n \times n$ matrix and ${\bf Q}$ is an invertible $m \times m$ matrix.