Question

A matrix a is given. determine if the system ax = b (where x and b have the appropriate number of components) has a solution for all choices of
b. a = 5 −4 4 2

Answer 1

We can write the system in the following form:
$\left[\begin{array}{cccc}5&-4&4&2\end{array}\right] \left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right] =b$
The above system is equivalent to the following equation:
$5x_1-4x_2+4x_3+2x_4=b,\forall b\in\mathbb{R}$
Of course, the above system has solution for any values of b since there is one equation and four variable, there is infinite number of solution each time.