Find conditions on α and β that ensure that Null(A) = {0} and Find conditions on α and β that ensure that Col(B) = R 3

Question

Let A and B be the matrices

$$ A= \begin{pmatrix} 1 & 1 & α \\ 1 & 0 & 0 \\ 0 & 2 & β \\ \end{pmatrix} ,B = \begin{pmatrix} 1 & 1 & β \\ α & 0 & 0 \\ 0 & 3 & α \\ \end{pmatrix} $$

I've managed to reduce both matrices to its echelon form:

$$ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & α \\ 0 & 0 & β-2α \\ \end{pmatrix} $$
and

$$ B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & β \\ 0 & 0 & α-3β \\ \end{pmatrix} $$

However, I can't seem to form a linear system of equations with α and β separate from each other so I can find values for them.

Any kind of help is appreciated!! Thank you

Answer 1

Guide:

To make sure that the nullity only consist of the zero vector, make sure that $A$ is nonsingular.

To make sure that the column space of $B$ is $\mathbb{R}^3$, also make sure that $B$ is nonsingular.

That is we require their RREF to be nonsingular, or equivalently, every column has a pivot element.

Hint:

I expect to see "$\ne 0$" in your condition. There are two conditions and you are not suppose to find their numerical values.