# 106 年 - 106 國立中山大學_碩士班招生考試_資工系(乙組)：工程數學#105789

【非選題】
1.
1.If A is an n x n matrix, then A is called idempotent if A2 = A. Let A and B be n x n idempotent matrices.

【題組】 1.1 Show that AB is idempotent if AB = BA.

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2.【題組】

1.2 Show that if A is idempotent, then is idempotent.

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4.【題組】1.4Find all values of k for which kA is also idempotent.

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5.

2. A periodic signal x(t) with a period T0= 10, 0 ≤t ≤ 10 by the equation

【題組】2.1 Sketch the periodic function x(t) over the time interval -10 s t ≤ 20.

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6.【題組】2.2 Determine the DC coefficient of the Fourier series, ao.

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7.【題組】2.3 Use the Fourier analysis integral to find al, the first Fourier series coefficient.

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8.【題組】

2.4 If we add a constant valie of one to x(t), we obtain the signal y(t) = 1 + x(1) with y(t) given over
one period by
The signal can be represented by a Fourier series, but with different coefficients:
Explain how bo and b! are related to ao and a . Note: you should not have to evaluate any new integrals

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9.

3. The matrix is

【題組】3.1 Find the characteristic polynomial.

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10.【題組】3.2 Find the eigenvalues,

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11.【題組】3.3 And Find the associated eigenvectors.

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12.4. Using Laplace transform and showing the details of your work, solve the initial value problem. y1'=-2y1+3y2, y2=4y1-y2, y1(0)=4, y2(0)=3,

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13.5.Find the general solution of the following differential equation.
y"+4y'+4y=sin2x

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14.6. Solve the following initial value problem.
y'"-2y"+4y'-8y=0,y(0)=-1, y'(0)=30, y"(0)=28,