This question occurs to me while I am self-studying machine learning. Let's assume there is a sequence of non-singular matrices $\{A_n\}$ whose size is the same and whose determinant goes to infinity: $$\lim\limits_{N\to\infty}|A_n|=+\infty.$$
Consider the sequence of quadratic forms of corresponding matrix inverses: $$\{q_n\buildrel \wedge \over =a^TA_n^{-1}a\}$$ where the vector $a$ is a general non-zero constant vector.
Do we have that this sequence vanishes in the limit $$\lim\limits_{N\to\infty}q_n=0?$$
I have an inclination to agree with it, but my math background is not strong to prove it. If, in the end, it is not true in general, what additional condition do we need to establish this limit? Thank you.