Given, 

The matrix $\mathrm{A}=\left[\begin{array}{ccc}0& 0& -1\\ 3& 4& 5\\ -2& -4& -7\end{array}\right]$

The determinant of A, |A| = 0 + 0 – 1( – 12 + 8) = 4 ≠ 0

This indicates that the matrix have inverse

In order to find adjoint of the matrix A let's find the minors of the matrix A $=\left[\begin{array}{ccc}\left|\begin{array}{cc}4& 5\\ -4& -7\end{array}\right|& \left|\begin{array}{cc}3& 5\\ -2& -7\end{array}\right|& \left|\begin{array}{cc}3& 4\\ -2& -4\end{array}\right|\\ \left|\begin{array}{cc}0& -1\\ -4& -7\end{array}\right|& \left|\begin{array}{cc}0& -1\\ -2& -7\end{array}\right|& \left|\begin{array}{cc}0& 0\\ -2& -4\end{array}\right|\\ \left|\begin{array}{cc}0& -1\\ 4& 5\end{array}\right|& \left|\begin{array}{cc}0& -1\\ 3& 5\end{array}\right|& \left|\begin{array}{ll}0& 0\\ 3& 4\end{array}\right|\end{array}\right]=\left[\begin{array}{ccc}-8& -11& -4\\ -4& -2& 0\\ 4& 3& 0\end{array}\right]$

Co-factors

$=\left[\begin{array}{ccc}(-1{)}^{1+1}(-8)& (-1{)}^{1+2}(-11)& (-1{)}^{1+3}(-4)\\ (-1{)}^{2+1}(-4)& (-1{)}^{2+2}(-2)& (-1{)}^{2+3}(0)\\ (-1{)}^{3+1}(4)& (-1{)}^{3+2}(3)& (-1{)}^{3+3}(0)\end{array}\right]=\left[\begin{array}{ccc}-8& 11& -4\\ 4& -2& 0\\ 4& -3& 0\end{array}\right]$

Transpose of the matrix A with co-factors

${{\mathrm{C}}_{\mathrm{A}}}^{\mathrm{T}}={\left[\begin{array}{ccc}-8& 11& -4\\ 4& -2& 0\\ 4& -3& 0\end{array}\right]}^{\mathrm{T}}=\left[\begin{array}{ccc}-8& 4& 4\\ 11& -2& -3\\ -4& 0& 0\end{array}\right]=\mathrm{adj}\mathrm{A}$

The inverse of the matrix $\mathrm{A},{\mathrm{A}}^{-1}=\frac{1}{\left|\mathrm{A}\right|}\mathrm{adj}\mathrm{A}$

$=\frac{1}{4}\left[\begin{array}{ccc}-8& 4& 4\\ 11& -2& -3\\ -4& 0& 0\end{array}\right]$