# Basis of vectors with a variable [duplicate]

So I have these bunch of matrices I want to find the value of variable "a" to find the basis for M2x2 using determinant test

$$\begin{pmatrix} 2 & 2 \\ 1 & -2 \\ \end{pmatrix}$$

$$\begin{pmatrix} 0 & 0 \\ 1 & 1 \\ \end{pmatrix}$$

$$\begin{pmatrix} 1 & a \\ 2 & -2 \\ \end{pmatrix}$$

$$\begin{pmatrix} 1 & a \\ 1 & -1 \\ \end{pmatrix}$$

What I could do is write them in a different way

$$A\pmatrix{2\\2\\1\\-2}+B\pmatrix{0\\0\\1\\1}+C\pmatrix{1\\a\\2\\-2}+D\pmatrix{1\\a\\1\\-1}$$

Now I can find the RREF but since those are letter "A"s i dont know what to do. Someone told me I should use the determinant test. How do I use the determinant test in this situation?

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## marked as duplicate by user1551, Ayman Hourieh, Stefan Hansen, Michael Albanese, Davide GiraudoFeb 1 '13 at 12:51

$det \begin{pmatrix} 2 & 0 & 1 & 1 \\ 2 & 0 & a & a \\ 1 & 1 & 2 & 1 \\ -2 & 1 & -2 & -1 \end{pmatrix} \neq 0$.