Quiz Chapter 09: Adequacy of Solutions

1. The row sum norm of the matrix \left[ A \right] = \begin{bmatrix} 6&-7&3&13 \\ 19&-21&23&-29 \\ 41&47&-51&61 \\ \end{bmatrix} is

2. The adequacy of the solution of simultaneous linear equations \left[ A \right] \left[ X \right] = \left[ C \right] depends on

3. Given a set of equations in matrix form \left[ A \right] \left[ X \right] = \left[ C \right], \, \left \| A \right \| = 250, \, \left \| A^{-1} \right \| = 40 and \varepsilon_{mach} = 0.119 \times 10^{-6}, then the number of significant digits you can at least trust in the solutions are

4. The solution to a set of simultaneous linear equations

• • • \begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \\ \end{bmatrix} \begin{bmatrix} x_{1}\\x_{2}\\x_{3}\\ \end{bmatrix} = \begin{bmatrix} 44\\94\\138\\ \end{bmatrix}

is given as

• • • \begin{bmatrix} x_{1}\\x_{2}\\x_{3}\\ \end{bmatrix} = \begin{bmatrix} 2\\4\\7\\ \end{bmatrix}

The solution to another set of simultaneous linear equations is given by (note the coefficient matrix is the same as above)

• • • \begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \\ \end{bmatrix} \begin{bmatrix} x_{1}\\x_{2}\\x_{3}\\ \end{bmatrix} = \begin{bmatrix} 43.99\\93.98\\138.03\\ \end{bmatrix}

is given as

• • • \begin{bmatrix} x_{1}\\x_{2}\\x_{3}\\ \end{bmatrix} = \begin{bmatrix} 214.01\\-208.01\\60\\ \end{bmatrix}

Based on the row sum norm, the condition number of the coefficient matrix is greater than (choose the largest possible value)

5. The condition number of the n \times n identity matrix based on the row sum norm is

6. Let \left[ A \right] = \begin{bmatrix} 1&2+\delta \\ 2-\delta&1 \\ \end{bmatrix}. Based on the row sum norm and given that \delta \rightarrow 0, \, \delta > 0, the condition number of the matrix is