Assignment on Systems of linear Equations

1.

Write each system of linear equations in matrix form:

$$x + y = 1\hspace{1in}3x + y + z = 1$$

$$y - z = 1\hspace{1in}x - y - z = 0$$

$$x + z = 0$$

2.

Write a system of linear equations equivalent to the matrix equation

$\begin{pmatrix} 1&2&3&4\\ 5&6&7&8\\ 9&10&11&12 \end{pmatrix}\begin{pmatrix} w\\ x\\ y\\ z \end{pmatrix}= \begin{pmatrix} 13\\ 14\\ 15 \end{pmatrix}$

3.

Solve the systems in the first problem by a sequence of elementary operations applied to both sides of the equations.

4.

Row reduce the following matrices to echelon form and solve the associated matrix equations: $A \mathbf{x} = \mathbf{0}$.

$\begin{pmatrix} 1&2&0\\ 0&0&0\\ 0&0&1 \end{pmatrix}\begin{pmatrix} 1&0&0\\ 0&2... ...pmatrix}\begin{pmatrix} 1&2&3&4\\ 0&5&6&7\\ 0&0&8&9\\ 0&0&0&10 \end{pmatrix}$