神马作文网几个线性代数问题请问刘老师这几道题怎么解?
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几个线性代数问题
请问刘老师这几道题怎么解?
请问刘老师这几道题怎么解?
第二个题目没想出高招
下解法仅供参考
由题意, 一方面
A^(n+1) = AA^n =
2 1 1 2xn+yn xn xn
1 2 1 乘 xn 2xn+yn xn
1 1 2 xn xn 2xn+yn
=
6xn+2yn 5xn+yn 5xn+yn
5xn+yn 6xn+2yn 5xn+yn
5xn+yn 5xn+yn 6xn+2yn
另一方面 A^(n+1) =
2xn+1+yn+1 xn+1 xn+1
xn+1 2xn+1+yn+1 xn+1
xn+1 xn+1 2xn+1+yn+1
所以有
xn+1 = 5xn+yn
yn+1 = -4xn
所以 xn+1 = 5xn - 4xn-1
再由 A = A+0E, A^2=5A-4E
知 x1=1,x2=5; y1=0,y2=-4
x^2-5x+4 = (x-1)(x-4)
xn = c0 + c1*4^n
代入初始值
1 = c0 + 4c1
5 = c0 +16c1
解得 c1=1/3, c0=-1/3
所以 xn = (4^n - 1)/3
进而 yn = -4(4^(n-1) - 1)/3 = (4-4^n)/3
下解法仅供参考
由题意, 一方面
A^(n+1) = AA^n =
2 1 1 2xn+yn xn xn
1 2 1 乘 xn 2xn+yn xn
1 1 2 xn xn 2xn+yn
=
6xn+2yn 5xn+yn 5xn+yn
5xn+yn 6xn+2yn 5xn+yn
5xn+yn 5xn+yn 6xn+2yn
另一方面 A^(n+1) =
2xn+1+yn+1 xn+1 xn+1
xn+1 2xn+1+yn+1 xn+1
xn+1 xn+1 2xn+1+yn+1
所以有
xn+1 = 5xn+yn
yn+1 = -4xn
所以 xn+1 = 5xn - 4xn-1
再由 A = A+0E, A^2=5A-4E
知 x1=1,x2=5; y1=0,y2=-4
x^2-5x+4 = (x-1)(x-4)
xn = c0 + c1*4^n
代入初始值
1 = c0 + 4c1
5 = c0 +16c1
解得 c1=1/3, c0=-1/3
所以 xn = (4^n - 1)/3
进而 yn = -4(4^(n-1) - 1)/3 = (4-4^n)/3