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    Ok so the question is:

    Let A be an nxn matrix where n \geq 2. Show A* = 0 (A* is the adjoint matrix) if and only if rank(A)\leq n-2.

    I'm really struggling with answering this. These are my thoughts:

    If rank(A) \leq n-2 then A contains at least 2 dependent rows. Does this automatically make A* = 0?

    If A*=0 then (-1)i+jdet(Aij) = 0 for all i and j. So det(Aij) = 0 for all i and j. Having a zero determinant implies dependence?

    I think you mean the adjugate matrix. I don't know the proof of the result, but I can imagine that it is true - almost every minor will obviously have 2 dependent rows or columns; the ones that don't obviously have dependent rows or columns lie on the diagonal.

    Yes, if the determinant of a matrix is zero then there is a dependent row or column.
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