Assignment on Systems of linear Equations

1. Write each system of linear equations in matrix form:

x + y = 1  3x + y + z= 1
y − z = 1  x − y − z = 0
x + z = 0

2. Write a system of linear equations equivalent to the matrix equation

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

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: Ax = 0.

(\begin{matrix}1 & 2 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 1 & 1 \end{matrix}) (\begin{matrix}1 & 2 & 0 & 0 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 3 & 9 \end{matrix}) (\begin{matrix}1 & 2 & 3 & 4 \\ 0 & 5 & 6 & 7 \\ 0 & 0 & 8 & 9 \\ 0 & 0 & 0 & 10 \end{matrix})