1) If matrix $\ A$ is of a size of $\ n (n>1)$, $\ A^*$ is it's adjugate, when $\ r_A < n - 1$, $\ A^*=$ _____.
2) If $\ A$ is a $\ 3X4 $ size matrix, $\ AA^T$ is ____ size symmetric matrix, $\ |A^TA|=$ _____.
In the first question I can only conduct that if $\ r_A < n - 1$ then there exist at least two rows of zeroes. But I don't really understand how it affects the adjugate. Something tells me that answer might be 0, but I don't really know how to show it.
In the second question the first blank is $\ 3$ obviously, but I don't know the answer for the second one.
1) $A^*$ is formed from the cofactors of $A$. On the other hand, $r_A<n-1$ implies that all its minors of order $n-1$ are $0$. Since the cofactors are those minors (save for a sign), $A^*=0$.
2) $r_A\leq3$, and the rank of a product is less or equal than the minimum of the ranks of the matrices involved, so $r_{A^TA}\leq3$. As $A^TA$ is a $4\times 4$ matrix, we get that $A^TA$ is not full rank, so $|A^TA|=0$.