How Does the Rank of a Matrix Influence Its Cofactor Matrix Becoming Zero?

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Homework Statement



Let A be an n x n matrix where n \geq 2. Show that A^{\alpha} = 0 (where A^{\alpha} is the cofactor matrix and 0 here denotes the zero matrix, whose entries are the number 0) if and only if rankA \leq n-2



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The Attempt at a Solution


No idea where to start with this, it's just an additional question in the lecture notes which I haven't gone through in tutorial. Thanks.
 
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The cofactor matrix is obtained by deleting rows and columns and taking the determinant. Given the rank<=n-2, what about the rank after deletion? What about the determinant?
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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