Consider the real matrices $A = \left( \begin{array} { c c c } m & 1 & 1 \\ 0 & m & 3 \end{array} \right)$ and $B = \left( \begin{array} { c c } 1 & m \\ 0 & m \\ 0 & 1 \end{array} \right)$. It is requested: a) ( 0.75 points) Study whether there exists some value of $m$ for which the matrix $B A$ has an inverse. b) ( 0.75 points) Study the rank of the matrix $A B$ as a function of the parameter $m$. c) (1 point) For $m = 1$, discuss the system $\left( A ^ { t } A \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a \\ a \\ a ^ { 2 } \end{array} \right)$ according to the values of $a$.
Consider the real matrices $A = \left( \begin{array} { c c c } m & 1 & 1 \\ 0 & m & 3 \end{array} \right)$ and $B = \left( \begin{array} { c c } 1 & m \\ 0 & m \\ 0 & 1 \end{array} \right)$. It is requested:
a) ( 0.75 points) Study whether there exists some value of $m$ for which the matrix $B A$ has an inverse.
b) ( 0.75 points) Study the rank of the matrix $A B$ as a function of the parameter $m$.
c) (1 point) For $m = 1$, discuss the system $\left( A ^ { t } A \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a \\ a \\ a ^ { 2 } \end{array} \right)$ according to the values of $a$.