Exercise 12

Consider the matrices $$\begin{aligned} A = \left[\begin{array}{rr} 0 & 1 \\ 1 & 0 \\ \end{array}\right], \qquad & B = \left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array}\right], \\ \\ C = \left[\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{array}\right], \qquad & D = \left[\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array}\right] \end{aligned}$$

1. Use row (and/or column) operations to find the determinants of these matrices.

2. Write the $6\times6$ and $7\times7$ matrices that follow in this pattern and state their determinants based on what you have seen.