所屬科目：【阿摩】未分類題庫
1. (10%) Find the inverse of the matrix A = $$\begin{bmatrix} 2 & 0 & 5 \\ 0 & 3 & 0 \\ 1 & 0 & 3 \end{bmatrix}$$.
2. (10%) Find the dimension of the space spanned by the three given vectors: $x^2, x^2 - x - 1$, and x + 1.
3. (10%) Let D be the differentiation operator on P3 (the set of all polynomials of degree less than 3). If the matrix B represents D with respect to $[1, x, x^2]$, and the matrix C represents D with respect to $[1, 2x, 4x^2 - 2]$. Verify that B is similar to C.
4. (10%) Find the local maxima of the function: $f(x, y) = \left(\frac{1}{3}x^3 + xy^2 - 4xy + 1\right)$
5. (10%) Compute the Gram-Schmidt QR factorization of the matrix: $$\mathbf{E} = \begin{bmatrix} 1 & -1 & 4 \\ 1 & 4 & -2 \\ 1 & 4 & 2 \\ 1 & -1 & 0 \end{bmatrix}$$
6. (10%) A derangement is a permutation of objects that leaves no object in the original position. For example, the permutation 21453 is a derangement of 12345 because no digit is left in its original position. List all derangements of 1234.
7. (20%) Consider an RSA system with public key (n, e) = (667, 137). First, find prime numbers p and q such that n = pq. Second, find the value of d in the private key (n, d). Third, answer the question "Is it possible to find more than one values of d in the range 0 < d < (p - 1)(q - 1) that works in decryption?" If your answer is YES, find all possible values of d; if your answer is NO, please give a brief proof.
8. (20%) Consider all possible binary relations on a set A with 4 elements. Answer the following 5 questions. Q1: How many binary relations are there? Q2: How many binary relations are reflexive? Q3: How many binary relations are symmetric? Q4: How many binary relations are anti-symmetric? Q5: How many binary relations are reflexive and symmetric and anti-symmetric simultaneously?
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