課程名稱︰化學數學二 課程性質︰化學系選修 課程教師︰陳振中 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰2011.04.27 考試時限（分鐘）：120 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Section A (60%) 1.(10%) Write down the echelon form of the matrix below and calculate the determinant. ┌1 4 2 3┐ │2 3 4 1│ │4 1 3 2│ └3 2 1 4┘ 2.(15%) Calculate exp(A) for A = ┌3 1┐ └2 4┘ (Hint:exp(A)=1+A+A^2/2!+...) 3.(20%) Determine whether the inverse of the following matrices exist. If so, determine the inverse. ┌1 4 7┐ │2 5 8│ └3 6 9┘ and ┌ 1 0 -1┐ │-2 3 0│ └ 1 -3 2┘ 4.(15%) Determine the general solution of x ┌X1┐ ┌1 0 1 0 1┐│X2│ │0 2 0 2 4││X3│ └1 0 2 1 0┘│X4│ └X5┘ Section B (40%) 1.(12%)Write down true(T) or false(F) for the following statements: a. Zero is NOT allowed as an eigenvalue of a matrix. b. Every nonzero column vector wuth n elements is an eigenvector of the unit matrix of order n. c. Every square matrix has an inverse. d. For a square matrix all of whose elements below the principal diagonal are zero, the eigenvalues are equal to the diagonal elements. e. The eigenvalues of diagonal matrix are equal to the diagonal elements. f. Enery nonzero linear conbination of two eigenvectorsthat correspond to the same eigenvalue of a matrix is an eigenvector of that matrix. 2.(10%)For the point group of C3v:[E,C13 C23 σv σv' σv"], prove that the product of any two different reflections will be equivalent to either C13 or C23 3.(8%)For the clockwise rotation by 2π/3 about a C3 axis along the (0,0,1) direction, the matrix respresentation of the operation of C13 is written as ┌cos(2π/3) -sin(2π/3 0┐ │sin(2π/3 cos(2π/3) 0│ └ 0 0 1┘ (a)Find the eigenvalues and normalized eigenvectors of the above matrix. (b)Write down the diagonal form of the matrix respresentation of C13 (c)Write down the linear combination of x,y and z which can diagonalize C13. 4.(10%)Determine the point groups of the following molecules: 1. ┌══┐ │ │ ├══┤ │ │ └══┘ 2. H2C=C=CH2 3. chair form of cyclohexane 4. boat form of cyclohexane 5. --

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