An investor has invested in three types of bonds: Corporate bonds, Government bonds, and Municipal bonds. Let $x,y,$ and $z$ represent the amounts invested in thousands of dollars in each type of bond, respectively. The investor has the following constraints:

The total investment amount is $$300,000.$

The investor wants to allocate twice as much money in Corporate bonds as in Municipal bonds.

The investment in Government bonds should be $$50,000$ more than in Corporate bonds.

$($a$)$ Generate the system of linear equations representing the above constraints.

$($b$)$ Solve, using matrix algebra, the system of equations from Question $1$ to find the amounts invested in each type of bond.