Chapter 10 - Section 10.6 - Determinants and Cramer's Rule - 10.6 Exercises - Page 743: 40

$1024$

Step 1. Write this determinant as: $\begin{array}( \\|A|= \\ \\ \end{array} \begin{vmatrix} 2 &0&0 &0&0 &0&0 &0&0 &0\\ 0 &2&0 &0&0 &0&0 &0&0 &0\\ 0 &0&2 &0&0 &0&0 &0&0 &0\\ 0 &0&0 &2&0 &0&0 &0&0 &0\\ 0 &0&0 &0&2 &0&0 &0&0 &0\\ 0 &0&0 &0&0 &2&0 &0&0 &0\\ 0 &0&0 &0&0 &0&2 &0&0 &0\\ 0 &0&0 &0&0 &0&0 &2&0 &0\\ 0 &0&0 &0&0 &0&0 &0&2 &0 \\ 0 &0&0 &0&0 &0&0 &0&0 &2 \end{vmatrix} $ Step 2. Use row 1 to expand the determinant: $\begin{array}( \\|A|=(2) \\ \\ \end{array} \begin{vmatrix} 2&0 &0&0 &0&0 &0&0 &0\\ 0&2 &0&0 &0&0 &0&0 &0\\ 0&0 &2&0 &0&0 &0&0 &0\\ 0&0 &0&2 &0&0 &0&0 &0\\ 0 &0 &0&0 &2&0 &0&0 &0\\ 0 &0 &0&0 &0&2 &0&0 &0\\ 0 &0 &0&0 &0&0 &2&0 &0\\ 0 &0 &0&0 &0&0 &0&2 &0 \\ 0 &0 &0&0 &0&0 &0&0 &2 \end{vmatrix} $ Step 3. Repeat step 2 nine more times, we can get: $|A|=(2)^{10}=1024$