考虑下面的2 × 2矩阵A：
其中两个元素是未知的，用a和b标记。这个矩阵的特征值是-1和7。a和b的值是多少？
a = 3, b = 5
a = 5, b = 3
a = 4, b = 6
a = 6, b = 4
给定矩阵的特征方程为
| 1-λ 4 | = 0
| b a-λ|
(1-λ)*(a-λ) - 4b = 0
令λ = -1，
=> (1 - (-1)) * (a - (-1)) - 4b = 0
=> 2a + 2 - 4b = 0
=> 2b - a = 1
令λ = 7，
=> (1 - 7) * (a - 7) - 4b = 0
=> -6a + 42 - 4b = 0
=> 2b + 3a = 21
解上述两个方程，我们得到
A = 5, b = 3
| 1-λ 4 | = 0
| b a-λ|
(1-λ)*(a-λ) - 4b = 0
令λ = -1，
=> (1 - (-1)) * (a - (-1)) - 4b = 0
=> 2a + 2 - 4b = 0
=> 2b - a = 1
令λ = 7，
=> (1 - 7) * (a - 7) - 4b = 0
=> -6a + 42 - 4b = 0
=> 2b + 3a = 21
解上述两个方程，我们得到
A = 5, b = 3
