Chapter 6.2, Problem 52E

Consider a $2\times 2$ matrix
   $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
with column vectors
   $\stackrel{\to }{v}=\left[\begin{array}{l}a\\ c\end{array}\right]$ and $\stackrel{\to }{w}=\left[\begin{array}{l}b\\ d\end{array}\right]$ .
We define the linear transformation
   $T\left(\stackrel{\to }{x}\right)=\left[\begin{array}{l}\det \left[\begin{array}{cc}\stackrel{\to }{x}& \stackrel{\to }{w}\end{array}\right]\\ \det \left[\begin{array}{cc}\stackrel{\to }{v}& \stackrel{\to }{x}\end{array}\right]\end{array}\right]$
from ${ℝ}^2$ to ${ℝ}^2$ .
a. Find the standard matrix B of T. (Write the entries of B in terms of the entries a, b, c, d of A.)
b. What is the relationship between the determinants of A and B?
c. Show that BA is a scalar multiple of $I_2$ . What about AB?
d. If A is noninvertible (but nonzero), what is the relationship between the image of A and the kernel of B? What about the kernel of A and the image of B?
e. If A is invertible, what ¡s the relationship between B and $A^{-1}$ ?