Finding an entry in Adj(A), given matrix A.

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I'm doing a practice exam and I came up with a different answer than the key, I'm not sure where I went wrong.

the question: Find the (2,4) entry of the matrix Adj(A),

if A = $$\begin{matrix}1 & 1 & 0 & 2\\ 0 & 1 & 2 & 0\\ 0 & 2 & -2 & 2\\ 0 & 0 & 1 & 1\\ \end{matrix}$$

I eliminated row 2 and column 4 then found the determinant of the resulting matrix and ended up with an answer of 2....the key says the answer is 4.

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1 Answer

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You have to take the minor $\;(-1)^{2+4}A_{42}\;$ , which is

$$\det\begin{pmatrix}1&0&2\\0&2&0\\0&\!-2&2\end{pmatrix}=4$$

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