Chapter 6.1, Problem 63E

In Exercises 62 through 64, consider a function D from ${ℝ}^{2\times 2}$2 to $ℝ$that is linear in both columns and alternating on the columns. See Examples 4 and 6 and the subsequent discussions. Assume that $D\left(I_2\right)=1$.

63. Show that $D\left[\begin{array}{cc}a& b\\ 0& d\end{array}\right]=ad$ . Hint: Write $\left[\begin{array}{l}b\\ d\end{array}\right]=\left[\begin{array}{l}b\\ 0\end{array}\right]+\left[\begin{array}{l}0\\ d\end{array}\right]$ and use linearity in the second column: $D\left[\begin{array}{cc}a& b\\ 0& d\end{array}\right]=D\left[\begin{array}{cc}a& b\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}a& 0\\ 0& d\end{array}\right]=abD\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]+\cdots$ Use Exercise 62.