课程名称︰管理数学 课程性质︰必修 课程教师︰谢承熹 开课学院：管理学院 开课系所︰财金系 考试日期（年月日）︰2016/11/08 考试时限（分钟）：120分钟 试题 : 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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