Find $m$ so that $\mbox{rank} \left( A^{-1} \right) = 3$

Question

Let matrix $$A=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 3 & m \\ 3 & 4 & 2\end{bmatrix}$$ Find $m$ so that $\text{rank} \left(A^{-1}\right)=3$

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Please help me with this problem. Thanks so much.

Answer 1

The rank is the order of the largest non-singular matrix. If the rank of matrix $A^{-1} = 3$, $\det{A} \neq 0$ $$\because \det{A} = 4-4m \ne 0 \implies m \neq 1$$ If I have made some error, please feel free to correct me. Hope this helps!