Determine all values of $k$ for which the following matrices are linearly independent in $M_{22}$

Question

If we express these matrix vectors as an augmented matrix, we get a row of zeros. If take out this row of zeros we are left with a $3x3$ matrix, is this allowed? We can find values for which the determinant of the matrix is $0$, for which I got $k=-1$ and $k=2$. These values represent the values of $k$ for which the vectors would be linearly dependent.

Is this correct? If there is a row of zeros, can I take it out and find the determinant of the remaining matrix?

Answer 1

We would need $$k=c_1+3c_2$$ Then $$1=c_1(c_1+3c_2)+c_2$$ and $$1=-c_1+2c_2$$ so $$1=(2c_2-1)(5c_2-1)+c_2=10c_2^2-6c_2+1$$ Thus $$5c_2=3,\hspace{2mm}c_2={3\over 5},\hspace{2mm}c_1={1\over 5}$$ and $$k=2$$ or $$c_2=0,\hspace{2mm}c_1=-1,\hspace{2mm}k=-1$$ You are correct.

Answer 2

Yes. But in general you will not end up with a square matrix, so the appropriate concept is the rank of the matrix (i.e. the number of leading ones in the row-reduced form); for your vectors to be linearly independent it has to agree with the number of vectors.