※ 引述《PttFund (批踢踢基金)》之銘言：
: Let
: ( 0 1 0 1 )
: A = ( 1 0 1 0 ).
: ( 0 1 0 1 )
: ( 1 0 1 0 )
: Find
: (1) the determinant of A.
: (2) the characteristic polynomial of A.
| 0 1 0 1 |←┐
(1) det(A) = | 1 0 1 0 | │一樣
| 0 1 0 1 |←┘
| 1 0 1 0 |
= 0.
| x -1 0 -1 | | x-2 -1 0 -1 |
(2) char_A(x) = det(Ix - A) = | -1 x -1 0 | = | x-2 x -1 0 |
| 0 -1 x -1 | | x-2 -1 x -1 |
| -1 0 -1 x | | x-2 0 -1 x |
| 1 -1 0 -1 | | 1 -1 0 -1 |
= (x-2)| 1 x -1 0 | = (x-2)| 0 x+1 -1 1 |
| 1 -1 x -1 | | 0 0 x 0 |
| 1 0 -1 x | | 0 1 -1 x+1 |
| x+1 -1 1 |
= (x-2)| 0 x 0 | = (x-2)( x(x+1)^2-x )
| 1 -1 x+1 |
= x^2(x+2)(x-2).
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