In a $2\times2$ matrix, it is quite easy to see if the vectors lie on a plane or not. By vector, I mean the columns of the matrix. I usually determine if the numbers are of a certain multiple. From there, I can judge that the columns of the matrix lie on a plane and that means its determinant is zero and therefore not invertible. So for instance: $ \left |\begin{bmatrix} 2 & 4\\ 4 & 8 \end{bmatrix} \right | = 0 $ Both vectors $\begin{bmatrix} 2\\ 4 \end{bmatrix}$ and $\begin{bmatrix} 4\\ 8 \end{bmatrix}$ lie on the same plane because the second vector is just 2 times of the first vector.
But in a $3\times3$ or $N\times N$ matrix, how can I get to know if a matrix is invertible or not by observing it without going through to calculate its determinant? For example in a matrix say $\begin{bmatrix} 1 & 4 & 7\\ 2 & 5 & 8\\ 3 & 6 & 9 \end{bmatrix}$, the numbers are not of a common multiple of each other. But still, the determinant of this matrix gives zero, which means the matrix is not invertible. But I wouldn't have known this without going through the trouble to calculate its determinant value. Since the numbers are not of a multiple, I am also not sure if the columns of the matrix lie on the same plane or not.
Thanks for any help.