How do I find matrix B when adj(B) is given? (a) find matrix B such that $adj(B)=A$, with $A$ given by :
$$A:=\left(\begin{array}{rrrr}
        1 & 2 & 5 & 4 \\
        0 & -1 & -2 & -1 \\
        -1 & 1 & 3 & 0 \\
        0 & 2 & 5 & 3 \\
        \end{array}\right)$$
(b) For the same matrix $A$, find all complex matrices $B$ such that $adj(B)=A$

Building up on Bye_World's comment, note that you can use the fact - if $A$ is $n×n$, then $\lvert (adj(A))\rvert = \lvert A\rvert^{n−1}$ to find the determinant of B and once you have the determinant of $B$ just plug it in your inverse formula 
$$B^{-1}=\frac{1}{\lvert B \rvert}adj(B)$$
to get $B^{-1}$ and then find its inverse
$$B=(B^{-1})^{-1}=\frac{1}{\lvert B^{-1} \rvert}adj(B^{-1})$$
to get $B$.
