107年 - 107 國立臺北大學_日間學士班曁進修學士班轉學生招生考試試題_資訊工程學系日間學士班3年級：線性代數#147393

1. (10%) Use the elementary row operations to find the inverse of the matrix \( \begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 1 \\ -1 & -2 & -3 \end{bmatrix} \)

2. (10%) Let \( A = \begin{bmatrix} 2 & 1 & 1 \\ 6 & 4 & 5 \\ 4 & 1 & 3 \end{bmatrix} \). Find an upper triangular matrix \( U \) and a lower triangular matrix \( L \) such that \( A = LU \).

3. (10%) The following matrix is the adjoint matrix of \( A \): adj \( A = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 4 & 3 & 2 \\ 0 & -2 & -1 & 2 \end{bmatrix} \). Please calculate \( \det(A) \).

(a)(10%) Find all solutions to the system.

(b)(10%) If $$ a_1 = \begin{bmatrix} 1 \\ 1 \\ 3 \\ 4 \end{bmatrix} $ and  a_3 = \begin{bmatrix} 2 \\ -1 \\ 1 \\ 3 \end{bmatrix} $ determine b

(a)(5%) Show that the mapping \( L \) is a linear transformation.

(b)(5%) Find a matrix \( \mathbf{A} \) such that \( L(\mathbf{x}) = \mathbf{A}\mathbf{x} \) for each \( \mathbf{x} \in \mathbb{R}^3 \).

(a)(10%) Find the least squares solution \( \hat{\mathbf{x}} \) for the system \( \mathbf{B}\mathbf{x} = \mathbf{b} \).

(b)(5%) Determine the projection \( \mathbf{p} = \mathbf{B}\hat{\mathbf{x}} \)

(c)(5%) Calculate the residual \( r(\hat{\mathbf{x}}) \)

(a)(10%) Find the eigenvalues and the corresponding eigenvectors for the matrix \( \mathbf{C} \).

(b)(5%) Factor the matrix \( \mathbf{C} \) into a product \( \mathbf{XDX}^{-1} \), where \( \mathbf{D} \) is diagonal.

(c)(5%) Use the \( \mathbf{XDX}^{-1} \) factorization in (b) to compute \( \mathbf{C}^{100} \).