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I am doing an exercise book which has one problem that asks you to prove the nonsingularity of a matrix if each element of the matrix equals its cofactor (the determinant submatrix by deleting the element's located row and column).

I've been thinking on this problem for two days but got no clue. My problem is I can't connect the conditions to any property the matrix should have. Anyone can give me some suggestion on how to prove this problem? And also how to think about the problem nicely?

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You have several questions with answers. Perhaps you'd like to accept some of those answers (hit the checkmark next to the answer)? Knowing that you reward those people who give good answers can be an incentive for others to look at your questions. – Muphrid Nov 20 '12 at 17:40

This is not true for the zero matrix. For non-zero matrices, use cofactor expansion to show that the determinant is non-zero. (We have taken cofactor to mean the signed minor here.)

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