[試題] 105-1 謝承熹 管理數學 期中考
課程名稱︰管理數學
課程性質︰必修
課程教師︰謝承熹
開課學院：管理學院
開課系所︰財金系
考試日期（年月日）︰2016/11/08
考試時限（分鐘）：120分鐘
試題 :
I (72 points) Answer each of the following as True or False. Justify your answer
, otherwise you cannot get any point. Moreover, please give your answer
in order. Thanks!
For Q1~Q10, let A and B be nxn matrices.
1. If A and B are scalar matrices, then AB is a scalar matrix.
2. If A and B are symmetric, then AB is symmetric.
3. If A and B are upper triangular, then AB is upper triangular.
4. If A and B are skew symmetric, then A+B is skew symmetric.
5. If A and B are in reduced row echelon form, then A+B is in reduced row
echelon form.
6. If A and B are nonsingular, then A+B is nonsingular.
7. If A and B are singular, then A+B is singular.
8. If A and B are idempotent, then A+B is idempotent.
9. If Trace(A) = Trace(B) = 1, then Trace(A+B) = 1.
10.If det(A) = det(B) = 1, then det(A+B) = 1.
11.If c and d are solutions of Ax=0, then c+d is a solution of Ax=0.
12.If c and d are solutions of Ax=b, then c+d is a solution of Ax=b.
II (14 points) Let A = [ai,j], where ai,j = xi^(j-1) for i = 1,2,3,4 and
j = 1,2,3,4. Find det(A).
III(10 points)(a) Let L:R2-->R2 be a linear transformation which projects any
2x1 vector into the vector span by (a b)2x1 orthogonally. Find the standard
matrix representing L. You can use the formula of projection matrix if necessary.
(4 points)(b) Base on the result of (a), verify the matrix is idempotent.
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