Let $\lambda$ be a real number and consider the matrices $A = \left( \begin{array} { c c c } \lambda & 1 & \lambda \\ 0 & \lambda & - 1 \end{array} \right)$ and $B = \left( \begin{array} { c c } 1 & \lambda \\ 0 & - 1 \\ 1 & - \lambda \end{array} \right)$. It is requested: a) ( 0.5 points) Determine whether there exists some value of $\lambda$ for which the matrix $AB$ does not have an inverse. b) (1 point) Study the rank of the matrix $BA$ as a function of the parameter $\lambda$. c) (1 point) For $\lambda = 1$, discuss the system $\left( A ^ { t } A \right) \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a ^ { 2 } \\ a ^ { 2 } \\ 2 a \end{array} \right)$ according to the values of $a$.
Let $\lambda$ be a real number and consider the matrices $A = \left( \begin{array} { c c c } \lambda & 1 & \lambda \\ 0 & \lambda & - 1 \end{array} \right)$ and $B = \left( \begin{array} { c c } 1 & \lambda \\ 0 & - 1 \\ 1 & - \lambda \end{array} \right)$. It is requested:
a) ( 0.5 points) Determine whether there exists some value of $\lambda$ for which the matrix $AB$ does not have an inverse.
b) (1 point) Study the rank of the matrix $BA$ as a function of the parameter $\lambda$.
c) (1 point) For $\lambda = 1$, discuss the system $\left( A ^ { t } A \right) \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a ^ { 2 } \\ a ^ { 2 } \\ 2 a \end{array} \right)$ according to the values of $a$.