課程名稱︰線性代數一
課程性質︰數學系大一必修
課程教師︰余正道
開課學院：理學院
開課系所︰數學系
考試日期︰2019年11月29日(五)，11:20-11:50
考試時限：30分鐘
試題 :
[Quiz 3]
1. We say that a matrix A∈M_n(C) is skew-symmetric if A^t = -A.
(a) (5%) Show that there is no invertible skew-symmetric matrix A∈M_n(C)
if n is odd.
(b) (5%) If A∈M_(2n)(C) is an invertible skew-symmetric matrix, show that
A^(-1) is also skew-symmetric.
2. Let V = M_2(R) be a 4-dimensional vector space over R and let a,b∈R.
Consider a linear operator on V defined by
T(B) = BA - AB where A = (a b)
(b a)
(a) (4%) Find all characteristic values of T.
(b) (4%) Find the characteristic spaces of each characteristic values.
(c) (2%) Find all a,b∈R such that T is diagonalizable.
3. (10%) Let n be a positive integer and A∈M_n(C) be a matrix such that
characteristic polynomial of A with real coefficients. Suppose that
A^4 - 2A^3 + A^2 + 2A - 2I = 0. Show that det(A) > 0 if and only if
n - tr(A) is divisible by 4.