Please could someone advise me on where to even start with this question. Thanks in advance
Let $A$ be an $n\times n$ matrix, for some $n\geqslant 1$. Given any prime number $p\gt1$, write $A_p$ for the matrix you get by replacing each entry in $A$ with its value ${\rm mod}\, p.$ For example, $$\pmatrix{ 1 & 27 & 11 \\ 2 & 10 & -3 \\ 0 & -4 & 17 }_3 = \pmatrix{ 1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 2 & 2 }$$ Define an operator $\odot_p$ ("matrix multiplication ${\rm mod}\,p\,$") by $A\odot_pB=(AB)_p.$ For example, $$\pmatrix{ 1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 2 & 1 \\ }\odot_3 \pmatrix{ 1 & 0 & 2 \\ 2 & 1 & 0 \\ 0 & 2 & 1 }=\pmatrix{ 1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 }_3=\pmatrix{ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 }$$ We'll say that $A$ is invertible-mod-p provided there exists some matrix $B$ such that $$A\odot_p B= B\odot_p A = I$$ where $I$ is the $n\times n$ identity matrix.
$a) \quad$ Prove by induction on $n\geqslant1$ that $|A_p|\equiv |A|\mod p$.
$b)\quad$ Using the result stated in $(a)$, prove that $A_p$ is invertible-mod-$p$ if and only if $|A|$ is not a multiple of $p$.