第14章 14.4

(1)

$\left|\begin{array}{cc}1 & 2\\3 & 4\end{array}\right| = 4-6=-2\neq 0 \qquad \text{よって正則}$

$\text{逆行列} = \frac{1}{-2}\begin{pmatrix}4 & -2\\-3 & 1\end{pmatrix} = \begin{pmatrix}-2 & 1\\ \dfrac{3}{2} & -\dfrac{1}{2}\end{pmatrix}$

(2)

$\left|\begin{array}{cc}4 & 6\\6 & 9\end{array}\right| = 36-36=0 \qquad \text{よって正則ではない}$

(3)

$\left|\begin{array}{cc}1 & 2\\-2 & 1\end{array}\right| = 1+4=5\neq 0 \qquad \text{よって正則}$

$\text{逆行列} = \frac{1}{5}\begin{pmatrix}1 & -2\\2 & 1\end{pmatrix} = \begin{pmatrix}\dfrac{1}{5} & -\dfrac{2}{5}\\ \dfrac{2}{5} & \dfrac{1}{5}\end{pmatrix}$

(4)

$\left|\begin{array}{cc}1 & a\\0 & 1\end{array}\right| = 1\neq 0 \qquad \text{よって正則}$

$\text{逆行列} = \begin{pmatrix}1 & -a\\0 & 1\end{pmatrix}$

(5)

$\left|\begin{array}{cc}\cos\theta & \sin\theta\\ \sin\theta & -\cos\theta\end{array}\right| = -\cos^2\theta-\sin^2\theta=-1\neq 0 \qquad \text{よって正則}$

$\text{逆行列} = -1\begin{pmatrix}-\cos\theta & -\sin\theta\\-\sin\theta & \cos\theta\end{pmatrix} = \begin{pmatrix}\cos\theta & \sin\theta\\ \sin\theta & -\cos\theta\end{pmatrix}$

(6)

$\left|\begin{array}{cc}1+a & -a\\a & 1-a\end{array}\right| = (1+a)(1-a)+a^2 = 1-a^2+a^2=1 \qquad \text{よって正則}$

$\text{逆行列} = \begin{pmatrix}1-a & a\\-a & 1+a\end{pmatrix}$