課程名稱︰管理數學
課程性質︰財金系大二必修
課程教師︰李世欽
開課學院：管理學院
開課系所︰財金系
考試時間︰11/17 2006 9:10~12:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. For the following system of linear equations with coefficient matrix A:
｛2Ｘ1+Ｘ2+2Ｘ3=5｝
｛3Ｘ1+Ｘ2-Ｘ3=10｝
｛X1+2X3=15 ｝
(a) (point 5) Solve the linear system
(b) (point 5) Compute the inverse of A
2.(a) (5 points) Prove Cramer's rule.
(b) (10 points) Solve the linear system by Cramer's rule.
(c) (10 points) Solve the linear system by Gauss-Jordan reduction.
｛Ｘ1+Ｘ2+Ｘ3-2Ｘ4=-4｝
Linear system ｛2Ｘ2+Ｘ3+3Ｘ4=4 ｝
｛2Ｘ1+Ｘ2-Ｘ3+2Ｘ4=5｝
｛Ｘ1-Ｘ2+Ｘ4=4 ｝
3.(10 points) Find all possible 3x3 matrices X for which AX=0, where
┌ ┐
│ 1 -2 3 │
A = │-2 5 -6 │
│ 2 -3 6 │
└ ┘
4. A, B, C are matrices. I is an identity matrix. 0 is a zero matrix. Prove or
disprove (give a counter-example) the following statements
(a) (5 points) if AB=BA then (A+B)^2=A^2+2AB+B^2
(b) (5 points) if A^n=0 then A=0
(c) (5 points) if A^2=A then A=0 or A=I
(d) (5 points) if AB=AC then B=C
5.(5 points) (a) Show that if A is singular, then A(adj A)=0
(5 points) (b) Show that if A is singular, then adj A is singular.
(5 points) (c) Show that if A is an nxn matrix,
then det(adj A)=[det(A)]^(n-1)
6. (10 points) Show that if A, B, (A+B) are invertible matrices, then
A(A+B)^(-1)B = B(A+B)^(-1)A
[ Hint:(A^(-1)+B^(-1))^(-1) = A(A+B)^(-1)B ]
7. (10 points) Suppose that A is a square matrix with A=-A^T and that (I-A) is
invertible, define B as B=(I+A)(I-A)^(-1), show that B^(-1) = B^T; where I
is an identity matrix.
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