山梨大学 医工農学総合教育部 工学専攻 機械工学コース 2022年7月実施 前期募集 数学 問1 Author 祭音Myyura (Based on Miyake's answer refined with GPT 6 Astra)
Description 問1 次に示す 3 × 3 3\times3 3 × 3 A , X A,X A , X λ \lambda λ a , b , c , d , e , f , g , h , i a,b,c,d,e,f,g,h,i a , b , c , d , e , f , g , h , i A X = X A AX=XA A X = X A a , b , c , d , e , f , g , h , i a,b,c,d,e,f,g,h,i a , b , c , d , e , f , g , h , i
Problem 1 Let A A A X X X 3 × 3 3\times3 3 × 3
A = ( λ 0 1 0 λ 0 1 1 λ ) , X = ( a b c d e f g h i ) A=\begin{pmatrix}\lambda&0&1\\0&\lambda&0\\1&1&\lambda\end{pmatrix},\qquad
X=\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix} A = λ 0 1 0 λ 1 1 0 λ , X = a d g b e h c f i where λ \lambda λ a , b , c , d , e , f , g , h , i a,b,c,d,e,f,g,h,i a , b , c , d , e , f , g , h , i a , b , c , d , e , f , g , h , i a,b,c,d,e,f,g,h,i a , b , c , d , e , f , g , h , i A X = X A AX=XA A X = X A
题目描述
问1 给定两个 3 × 3 3\times3 3 × 3
A = ( λ 0 1 0 λ 0 1 1 λ ) , X = ( a b c d e f g h i ) , A=\begin{pmatrix}\lambda&0&1\\0&\lambda&0\\1&1&\lambda\end{pmatrix},\qquad
X=\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}, A = λ 0 1 0 λ 1 1 0 λ , X = a d g b e h c f i , 其中 λ \lambda λ a , b , c , d , e , f , g , h , i a,b,c,d,e,f,g,h,i a , b , c , d , e , f , g , h , i A X = X A AX=XA A X = X A a , b , c , d , e , f , g , h , i a,b,c,d,e,f,g,h,i a , b , c , d , e , f , g , h , i
Kai A = λ I + N A=\lambda I+N A = λ I + N A X = X A AX=XA A X = X A N X = X N NX=XN NX = XN
N X − X N = ( g − c h − c i − a − f − f − d a + d − i b + e − i c + f − g ) . NX-XN=\begin{pmatrix}
g-c&h-c&i-a\\
-f&-f&-d\\
a+d-i&b+e-i&c+f-g
\end{pmatrix}. NX − XN = g − c − f a + d − i h − c − f b + e − i i − a − d c + f − g . したがって
d = f = 0 , g = h = c , i = a , e = a − b . d=f=0,\qquad g=h=c,\qquad i=a,\qquad e=a-b. d = f = 0 , g = h = c , i = a , e = a − b . a , b , c a,b,c a , b , c 0 0 0 d , f \boxed{d,f} d , f
Reference