generator for a vector space

Question

I stumbled upon an exercise where I had to determine a value k inside S :(A1,A2,A3), with A1,A2 and A3 all being 2x2 matrices, so that S generates a vector space. (k being atleast once in every matrix)

I normally solve to find the basis by checking linear independence and getting the value for k that way. But here since there are only 3 matrices inside S that are 2x2 I get a 4x3 matrix for which I can't check de determinant so I'm kinda clueless at this point how to solve for k.

sorry if my words don't seem logic, I'm not familiar with every English term for these kind of exercises.

Answer 1

Let $M$ be your $4 \times 3$ matrix. You want to know whether $M$ has full rank $3$. If you want to use determinants, you can look at $\det(M^TM)$. Indeed, $M^TM$ will be a $3 \times 3$ matrix which will be invertible if and only if $M$ has full rank $3$.

This follows from $\ker M = \ker (M^TM)$ so one matrix is injective if and only if the other one is.