# Chapter 16: Chapter 16

Problem 368

Solve the linear system $$ \begin{aligned} &10 \mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=15 \\ &\mathrm{x}_{1}+10 \mathrm{x}_{2}+\mathrm{x}_{3}=24 \\ &\mathrm{x}_{1}+\mathrm{x}_{2}+10 \mathrm{x}_{3}=33 \end{aligned} $$ using the Jacobi method.

Problem 369

Use the Gauss-Seidel method to solve the following linear system: $$ \begin{aligned} &10 \mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=15 \\ &\mathrm{x}_{1}+10 \mathrm{x}_{2}+\mathrm{x}_{3}=24 \\ &\mathrm{x}_{1}+\mathrm{x}_{2}+10 \mathrm{x}_{3}=33 \end{aligned} $$

Problem 370

Solve the linear system $$ \begin{aligned} &14 \mathrm{x}_{1}+2 \mathrm{x}_{2}+4 \mathrm{x}_{3}=-10 \\ &16 \mathrm{x}_{1}+40 \mathrm{x}_{2}-4 \mathrm{x}_{3}=55 \\ &-2 \mathrm{x}_{1}+4 \mathrm{x}_{2}-16 \mathrm{x}_{3}=-38 \end{aligned} $$ using the Gauss-Seidel iteration method.

Problem 372

Solve the following linear system using the method of relaxation: $$ \begin{aligned} 3 x+9 y-2 z &=11 \\ 4 x+2 y+13 z &=24 \\ 11 x-4 y+3 z &=-8 \end{aligned} $$

Problem 373

Find the smallest eigenvalue of the matrix $$ \mathrm{A}=\mid \begin{array}{rrr} 4 & 1 & -1 \mid \\ 2 & 3 & -1 \mid \\ -2 & 1 & 5 \mid \end{array} $$

Problem 375

Find the largest characteristic value of \(\lambda\) for the system $$ \begin{aligned} &\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=\lambda \mathrm{x}_{1} \\ &\mathrm{x}_{1}+2 \mathrm{x}_{2}+2 \mathrm{x}_{3}=\lambda \mathrm{x}_{2} \\ &\mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3}=\lambda \mathrm{x}_{3} \end{aligned} $$

Problem 377

The largest eigenvalue of the matrix $$ \mathrm{A}=\begin{array}{|rrr} \mid 0 & 5 & -6 \\ -4 & 12 & -12 \mid \\ \mid-2 & -2 & 10 \end{array} $$ was found, using the power method, to be \(\lambda_{1}=16\). An associated eigenvector was \(\mathrm{v}_{1}=[0.5,1.0,-0.5]\). Find the remaining eigenvalues and eigenvectors of \(\mathrm{A}\)

Problem 378

Let $$ \mathrm{A}=\begin{array}{rrr} 14 & 1 & 2 \\ 2 & 4 & -3 \mid \\ 3 & 1 & 3 \end{array} $$ Find the dominant root of A from the limiting form of a high power of the matrix.

Problem 379

Eigensystems arise in the physical sciences when we study vibrations. In such systems the eigenvector corresponding to the smallest eigenvalue will have elements that are all of the same sign. Using this information, estimate the smallest eigenvalue of the following matrix by means of the Rayleigh quotient: $$ \mathrm{A}=\mid \begin{array}{rrr} 1.7 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array} $$

Problem 380

Describe a matrix method for finding a) the cube root of 2 . b) the fourth root of 5 .