2025-10-10¶
Which of the following matrices is ill-conditioned?
\(\begin{bmatrix} 10^{10} & 0 \\ 0 & 10^{-10} \end{bmatrix}\)
\(\begin{bmatrix} 10^{10} & 0 \\ 0 & 10^{10} \end{bmatrix}\)
\(\begin{bmatrix} 10^{-10} & 0 \\ 0 & 10^{-10} \end{bmatrix}\)
\(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\)
Answer
a2. You are given a system of equations containing a real parameter \(a\). \(2 x_1 + 3 x_2 = 5, 4 x_1 + a x_2 = 8\) Which of these statements is true?
For \(a = 6\), the system has no solution
For \(a = 6\), the system has infinitely many solutions
The system has a unique solution for any real value of \(a\)
The two lines intersect when \(a = 6\)
Answer
a3. Suppose we are using a Krylov method to form an iterative solution \(x_n\) to the matrix equation \(A x = b\) from the basis vectors \(\left\lbrace b, A b, A^2 b, \dots \right\rbrace\). That is to say, \(x_n = c_0 b + c_1 A b + c_2 A^2 b + \cdots + c_n A^n b\). If \(A\) is an invertible \(m \times m\) matrix, what is the maximum number of vectors needed to find the solution \(A x_n = b\) and why?