高専の数学解説

第16章 16.20

(1)

∣123202321∣=12+12−4−4=16\begin{vmatrix} 1 & 2 & 3 \\ 2 & 0 & 2 \\ 3 & 2 & 1 \end{vmatrix} = 12 + 12 - 4 - 4 = 16

∣0221∣=−4∣2231∣=2−6=−4∣2032∣=4\begin{vmatrix} 0 & 2 \\ 2 & 1 \end{vmatrix} = -4 \quad \begin{vmatrix} 2 & 2 \\ 3 & 1 \end{vmatrix} = 2 - 6 = -4 \quad \begin{vmatrix} 2 & 0 \\ 3 & 2 \end{vmatrix} = 4

∣2321∣=2−6=−4∣1331∣=1−9=−8∣1232∣=2−6=−4\begin{vmatrix} 2 & 3 \\ 2 & 1 \end{vmatrix} = 2 - 6 = -4 \quad \begin{vmatrix} 1 & 3 \\ 3 & 1 \end{vmatrix} = 1 - 9 = -8 \quad \begin{vmatrix} 1 & 2 \\ 3 & 2 \end{vmatrix} = 2 - 6 = -4

∣2302∣=4∣1322∣=2−6=−4∣1220∣=−4\begin{vmatrix} 2 & 3 \\ 0 & 2 \end{vmatrix} = 4 \quad \begin{vmatrix} 1 & 3 \\ 2 & 2 \end{vmatrix} = 2 - 6 = -4 \quad \begin{vmatrix} 1 & 2 \\ 2 & 0 \end{vmatrix} = -4

これから

116(−4444−8444−4)=14(−1111−2111−1)\frac{1}{16}\begin{pmatrix} -4 & 4 & 4 \\ 4 & -8 & 4 \\ 4 & 4 & -4 \end{pmatrix} = \frac{1}{4}\begin{pmatrix} -1 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -1 \end{pmatrix}

(2)

∣1ab0−1c001∣=−1\begin{vmatrix} 1 & a & b \\ 0 & -1 & c \\ 0 & 0 & 1 \end{vmatrix} = -1

∣−1c01∣=−1∣0c01∣=0∣0c01∣=0\begin{vmatrix} -1 & c \\ 0 & 1 \end{vmatrix} = -1 \quad \begin{vmatrix} 0 & c \\ 0 & 1 \end{vmatrix} = 0 \quad \begin{vmatrix} 0 & c \\ 0 & 1 \end{vmatrix} = 0

∣ab01∣=a∣1b01∣=1∣1a00∣=0\begin{vmatrix} a & b \\ 0 & 1 \end{vmatrix} = a \quad \begin{vmatrix} 1 & b \\ 0 & 1 \end{vmatrix} = 1 \quad \begin{vmatrix} 1 & a \\ 0 & 0 \end{vmatrix} = 0

∣ab−1c∣=ac+b∣1b0c∣=c∣1a0−1∣=−1\begin{vmatrix} a & b \\ -1 & c \end{vmatrix} = ac + b \quad \begin{vmatrix} 1 & b \\ 0 & c \end{vmatrix} = c \quad \begin{vmatrix} 1 & a \\ 0 & -1 \end{vmatrix} = -1

これから

−(−1−aac+b01−c00−1)=(1a−ac−b0−1c001)-\begin{pmatrix} -1 & -a & ac+b \\ 0 & 1 & -c \\ 0 & 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & a & -ac-b \\ 0 & -1 & c \\ 0 & 0 & 1 \end{pmatrix}

(３)

∣1−100−12−100−12−100−13∣=∣1−10001−100−12−100−13∣=∣1−10−12−10−13∣=∣1−1−13∣=3−1=2\begin{vmatrix} 1 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 3 \end{vmatrix} = \begin{vmatrix} 1 & -1 & 0 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 3 \end{vmatrix} = \begin{vmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{vmatrix} = \begin{vmatrix} 1 & -1 \\ -1 & 3 \end{vmatrix} = 3 - 1 = 2

∣2−10−12−10−13∣=7∣−1−1002−10−13∣=−5∣−1200−1−1003∣=3∣−12−10−1200−1∣=−1\begin{vmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{vmatrix} = 7 \quad \begin{vmatrix} -1 & -1 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 3 \end{vmatrix} = -5 \quad \begin{vmatrix} -1 & 2 & 0 \\ 0 & -1 & -1 \\ 0 & 0 & 3 \end{vmatrix} = 3 \quad \begin{vmatrix} -1 & 2 & -1 \\ 0 & -1 & 2 \\ 0 & 0 & -1 \end{vmatrix} = -1

∣−100−12−10−13∣=−5∣10002−10−13∣=5∣1−10−1−1−1003∣=−3∣1−100−1200−1∣=1\begin{vmatrix} -1 & 0 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{vmatrix} = -5 \quad \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 3 \end{vmatrix} = 5 \quad \begin{vmatrix} 1 & -1 & 0 \\ -1 & -1 & -1 \\ 0 & 0 & 3 \end{vmatrix} = -3 \quad \begin{vmatrix} 1 & -1 & 0 \\ 0 & -1 & 2 \\ 0 & 0 & -1 \end{vmatrix} = 1

∣−1002−10003∣=3∣100−1−100−13∣=−3∣1−10−120003∣=3∣1−1−1−12−100−1∣=−1\begin{vmatrix} -1 & 0 & 0 \\ 2 & -1 & 0 \\ 0 & 0 & 3 \end{vmatrix} = 3 \quad \begin{vmatrix} 1 & 0 & 0 \\ -1 & -1 & 0 \\ 0 & -1 & 3 \end{vmatrix} = -3 \quad \begin{vmatrix} 1 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix} = 3 \quad \begin{vmatrix} 1 & -1 & -1 \\ -1 & 2 & -1 \\ 0 & 0 & -1 \end{vmatrix} = -1

∣−1002−10−12−1∣=−1∣100−1−1002−1∣=1∣1−10−1200−1−1∣=−1∣1−10−12−10−12∣=1\begin{vmatrix} -1 & 0 & 0 \\ 2 & -1 & 0 \\ -1 & 2 & -1 \end{vmatrix} = -1 \quad \begin{vmatrix} 1 & 0 & 0 \\ -1 & -1 & 0 \\ 0 & 2 & -1 \end{vmatrix} = 1 \quad \begin{vmatrix} 1 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & -1 & -1 \end{vmatrix} = -1 \quad \begin{vmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{vmatrix} = 1

これから

12(7531553133311111)\frac{1}{2}\begin{pmatrix} 7 & 5 & 3 & 1 \\ 5 & 5 & 3 & 1 \\ 3 & 3 & 3 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}

解説: r31bn1z