9/29 作业

Homework 6

1. (1) $2 \left| \begin{matrix} 6 & 3 \\ 4 & -7 \end{matrix} \right| - 1 \left| \begin{matrix} -3 & 4 \\ 4 & -7 \end{matrix} \right| + (-8) \left| \begin{matrix} -3 & 4 \\ 6 & 3 \end{matrix} \right| = 151$ .

   (2)

   (a) $(-3) \left| \begin{matrix} 4 & 7 & 1 \\ 5 & 10 & 5 \\ 0 & 5 & 0 \end{matrix} \right| = 225$ .

   (b) $- 1 \left| \begin{matrix} 3 & 0 & 0 \\ 8 & 5 & 10 \\ 6 & 0 & 5 \end{matrix} \right| - 5 \left| \begin{matrix} -2 & 4 & 7 \\ 3 & 0 & 0 \\ 6 & 0 & 5 \end{matrix} \right| = 225$ .

2. (1) $412$ .

   (2) $24$ .

3. (1) $x \in \left\{ 1, 2 \right\}$ .

   (2) $x \in \left\{ 2, 3 \right\}$ .

4. Proof By expanding $\square$ .

5. a. Proof

   Expand the determinant

   $$f(t) = (b - a) t^2 + (a^2 - b^2) t + ab^2 - a^2b$$

   Therefore $f(t)$ is quadratic function of $t$ $\square$ .

   The coefficient is $(b-a)$ .

   b. Apply $f$ , $f(a) = f(b) = 0$ .

   By factor theorem, $f(t) = k (t-a) (t-b)$ .

   $k = b - a$ .

   c. $\R - \left\{ a, b \right\}$ .