高専の数学解説

第18章 18.19

(1)

A=[f(e1)f(e2)f(e3)f(e4)]=[1111122212331234]A = \begin{bmatrix} f(e_1) \\ f(e_2) \\ f(e_3) \\ f(e_4) \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 \\ 1 & 2 & 3 & 3 \\ 1 & 2 & 3 & 4 \end{bmatrix}

(2)

∣A∣=∣1111122212331234∣=∣1111011101220123∣=∣111122123∣−∣111122123∣=∣1112∣=2−1=1|A| = \begin{vmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 \\ 1 & 2 & 3 & 3 \\ 1 & 2 & 3 & 4 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 2 & 2 \\ 0 & 1 & 2 & 3 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{vmatrix} - \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{vmatrix} = \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} = 2 - 1 = 1

(3)

A−1=1∣A∣[a11−a12a13−a14−a21a22−a23a24a31−a32a33−a34−a41a42−a43a44]=[a11−a12a13−a14−a21a22−a23a24a31−a32a33−a34−a41a42−a43a44]A^{-1} = \frac{1}{|A|} \begin{bmatrix} a_{11} & -a_{12} & a_{13} & -a_{14} \\ -a_{21} & a_{22} & -a_{23} & a_{24} \\ a_{31} & -a_{32} & a_{33} & -a_{34} \\ -a_{41} & a_{42} & -a_{43} & a_{44} \end{bmatrix} = \begin{bmatrix} a_{11} & -a_{12} & a_{13} & -a_{14} \\ -a_{21} & a_{22} & -a_{23} & a_{24} \\ a_{31} & -a_{32} & a_{33} & -a_{34} \\ -a_{41} & a_{42} & -a_{43} & a_{44} \end{bmatrix}

とおくと

a11=∣222233234∣=∣222011012∣=2∣1112∣=2(2−1)=2a_{11} = \begin{vmatrix} 2 & 2 & 2 \\ 2 & 3 & 3 \\ 2 & 3 & 4 \end{vmatrix} = \begin{vmatrix} 2 & 2 & 2 \\ 0 & 1 & 1 \\ 0 & 1 & 2 \end{vmatrix} = 2 \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} = 2(2-1) = 2

a12=∣111233234∣=∣111011012∣=∣1112∣=2−1=1a_{12} = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 3 & 3 \\ 2 & 3 & 4 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 2 \end{vmatrix} = \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} = 2 - 1 = 1

a13=∣111222234∣=∣111000012∣=0a_{13} = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 2 & 3 & 4 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 2 \end{vmatrix} = 0

a14=∣111222233∣=∣111000011∣=0a_{14} = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 2 & 3 & 3 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 1 \end{vmatrix} = 0

a22=∣111133134∣=∣111022023∣=∣2223∣=6−4=2a_{22} = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 3 & 3 \\ 1 & 3 & 4 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 2 & 3 \end{vmatrix} = \begin{vmatrix} 2 & 2 \\ 2 & 3 \end{vmatrix} = 6 - 4 = 2

a23=∣111122134∣=∣111011023∣=∣1123∣=3−2=1a_{23} = \begin{vmatrix}1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 4\end{vmatrix} = \begin{vmatrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 2 & 3\end{vmatrix} = \begin{vmatrix}1 & 1 \\ 2 & 3\end{vmatrix} = 3 - 2 = 1

a24=∣111122133∣=∣111011022∣=∣1122∣=2−2=0a_{24} = \begin{vmatrix}1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 3\end{vmatrix} = \begin{vmatrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 2 & 2\end{vmatrix} = \begin{vmatrix}1 & 1 \\ 2 & 2\end{vmatrix} = 2 - 2 = 0

a33=∣111122124∣=∣111011013∣=∣1113∣=3−1=2a_{33} = \begin{vmatrix}1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 4\end{vmatrix} = \begin{vmatrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 3\end{vmatrix} = \begin{vmatrix}1 & 1 \\ 1 & 3\end{vmatrix} = 3 - 1 = 2

a34=∣111122123∣=∣111011012∣=∣1112∣=2−1=1a_{34} = \begin{vmatrix}1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3\end{vmatrix} = \begin{vmatrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 2\end{vmatrix} = \begin{vmatrix}1 & 1 \\ 1 & 2\end{vmatrix} = 2 - 1 = 1

a44=∣111122123∣=∣111011012∣=∣1112∣=2−1=1a_{44} = \begin{vmatrix}1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3\end{vmatrix} = \begin{vmatrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 2\end{vmatrix} = \begin{vmatrix}1 & 1 \\ 1 & 2\end{vmatrix} = 2 - 1 = 1

これから

A=[2−100−12−100−12−100−11]A = \begin{bmatrix}2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 1\end{bmatrix}

解説: r31bn1z