課程名稱︰管理數學
課程性質︰財金系大二必修
課程教師︰謝承熹
開課系所︰財金系
考試時間︰2006/01/02 12:40～2:00
試題 :
1. If matrix Ａ is defined as follows, find sum of the matrix Ａ's eigenvalues.
(5 1 2 4)
(1 6 3 5)
Ａ=(2 3 7 6)
(4 5 6 8)
2. If matrix Ａ is defined as follows, find a matrix Ｐ such that Ｐ^(-1)ＡＰ
is a diagonal matrix.
(0 2 2)
Ａ=(2 0 2)
(2 2 0)
3. Let Ｓ=(v1, v2,....., vn) be a set of nonzero vectors in a vector space Ｖ
such that every vector in Ｖ can be written in one and only one way as a
linear combination of the vectors in Ｓ. Show that Ｓ is a basis for Ｖ.
4. Show that an n*n matrix Ａ is nonsingular if and only if rank Ａ=n.
5. Suppose that v1, v2,....., vn is an orthogonal set in Ｒ_n. Let Ａ be the
matrix whose jth column is vj, j=1,2,...,n. Prove or disprove: Ａ is
nonsingular.
6. Let λ be an eigenvalue of Ａ with associated eigenvector ｘ. Show that if
Ａ is nonsingular, then 1/λ+r is an eigenvalue of Ａ^(-1)+rＩn with
associated eigenvector ｘ.
7. Show that if Ａ is diagonalizable, then:(a) Ａ' is diagonalizable. (b) Ａ'
is diagonalizable, where k is a positive integer.
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