4. (10%) Answer the questions about functions. First, how many functions are there from a set with eight elements to a set with three elements? Second, how many onto functions are there from a set with eight elements to a set with three elements?

5. (10%) Let \( X = \{1, 2, 3, \ldots, 9\} \). Algorithm A lists all 6-permutations of \( X \) in increasing lexicographic order. Algorithm B prints all 6-combinations of \( X \) in increasing lexicographic order where each combination is printed as an increasing sequence. First, find the next five permutations that will be generated by algorithm A after the permutation 321978. Then, find the next five combinations that will be generated by algorithm B after the combination 123689.

6. (10%) Solve the recurrence relation \( a_n = (4 / n)(a_0 + a_1 + \dots + a_{n-1}) \) where \( a_0 = 3 \).

7. (20%) Given the graph shown below, answer the following questions and prove your answers. Does it have a Euler circuit? Does it have a Euler path? Does it have a Hamilton circuit? Does it have a Hamilton path?

26. (5%) Let \( \mathbf{U} \) be an \( n \times n \) unit upper triangular matrix. It is known that \( \mathbf{U} \) is nonsingular. Now, \( \mathbf{U}^{-1} \) is derived by transforming \( [\mathbf{U} \mathbf{I}] \) into \( [\mathbf{I} \mathbf{U}^{-1}] \) via row operations. Use the above derivation method to prove that \( \mathbf{U}^{-1} \) is also unit upper triangular.

(27) (10%) Let \( \mathbf{A} \) be a symmetric positive definite matrix with Cholesky decomposition \( \mathbf{A} = \mathbf{R}_{1}^{\text{T}}\mathbf{R}_{1} = \mathbf{R}_{2}^{\text{T}}\mathbf{R}_{2} \). By using the uniqueness of LDU factorization, prove that the upper triangular matrices \( \mathbf{R}_{1} = \mathbf{R}_{2} \).

1. If \( \mathbf{A}\mathbf{A}^{-1} = \mathbf{I} \), then \( \mathbf{A} \) must be equal to \( \mathbf{I} \) or \( -\mathbf{I} \). (I: the identity matrix)

2. The determinant of matrix \( \mathbf{A} \) is nonzero if and only if the reduced row echelon form of \( \mathbf{A} \) is the identity matrix.

3. A triangular matrix is singular if and only if one of its diagonal entries is zero.

4. If two augmented matrices \( (\mathbf{A}|\mathbf{b}) \) and \( (\mathbf{B}|\mathbf{c}) \) are row equivalent, then the determinants of \( \mathbf{A} \) and \( \mathbf{B} \) are equal.

5. \( \mathbf{R}^3 \) is a subspace of \( \mathbf{R}^5 \).

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