if $A^{-1}=(b_{ij})$, compute $b_{23}$ using cofactor.
$$ \begin{pmatrix} 1&2&3\\ 0&0&4\\ 5&0&1 \end{pmatrix} $$
I know matrix of cofactors can be obtained by transpose of matrix $A$ and computing cofactors.
But, how can I compute determinant of $A$ which is $3\times 3$ matrix? or Is there are other solutions that not to use determinant?
But the determinant is easy to compute: $\left |\begin{matrix}1&2&3\\0&0&4\\5&0&1\end{matrix}\right|=-4\cdot \left|\begin{matrix}1&2\\5&0\end{matrix}\right |=-4\cdot-10=40$.
(This was done by Laplace expansion, or expansion by minors, along the second row: you actually get $$-0\cdot A_{21}+0\cdot A_{22}-4\cdot A_{23}$$.)