10/9 作业

1. (1) $1$ , invertible.

   (2) $12$ , invertible.

   (3) $-12$ , invertible.

   (4) $8k$ , invertible iff. $k \ne 0$ .

   (5) $2k^2 - 2$ , invertible iff. $k \ne \pm 1$ .

   (6) $2$ , invertible.

   (7) $a^4 - 6a^2b^2 + 8ab^3 -3b^4$ , invertible iff. $a \ne b$ and $a \ne -3b$ .

   (8) $abcd$ , invertible iff. $abcd \ne 0$ .

   (9) $1$ , invertible.

2. (1) $\left\{ 2, 3, 5 \right\}$ .

   (2) $\left\{ 2, 3, 8 \right\}$ .

3. $(a,b,c,d,e,f,g,h) = (1,1,1,1,1,1,1,1)$ .

4. (1) $|AB| = -12$ , $|5A| = -375$ , $|B^\mathrm T| = 4$ , $|A^{-1}| = -\frac{1}{3}$ , $|A^3| = -27$ .

   (2) $|AB| = 3$ , $|B^5| = -1$ , $|2A| = -48$ , $|A^\mathrm T BA| = -9$ , $|B^{-1}AB| = -3$ .

5. (1) False.

   (2) True.