Exercises 45-54

Solve the system linear of equations

2x_{1 } | + | 7x_{2 } | + | x_{3 } | = | 19 | . | |

4x_{1 } | + | x_{2 } | - | x_{3 } | = | 3 | ||

x_{1 } | - | 3x_{2 } | + | 12x_{3 } | = | 31 | ||

In part a), compute the iteration matrix *G* and find out if it has norm smaller that 1
for some of the
matrix norms.

Hint: First change the order of equations to obtain a strictly diagonally dominant coefficient matrix.

The coefficient matrix of the linear system

2x + y + z = 4 | |

x + 2y + z = 4 | |

x + y + 2z = 4 | |

and compute iteration

Use the Gauss-Seidel method and solve the system

3x_{1 } +x_{3 } = 4 | |

-x_{1 } -x_{2 } +3x_{3 } = 1 | |

x_{1 } +2x_{2 } = 3 | |

Consider the over determined system

x_{1 } | - | x_{3 } | = | 4 |

x_{1 } | - | 3x_{3 } | = | 6 |

x_{2 } | + | x_{3 } | = | - 1 |

- x_{2 } | + | x_{3 } | = | 2 |

Verify, by straight-forward calculation, that the matrix

A = | 1 | -1 | 1 | ||

0 | 0 | 1 | |||

0 | 0 | 1 | |||

satisfies its own characteristic equation.
Use the Cayley-Hamilton theorem and compute *A ^{5}*.

Exercises 61-64

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