MHB Prove These 3 Statements About A Matrix: Rank of Adjoint

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SUMMARY

This discussion focuses on proving three statements regarding the rank of the adjugate of an nxn matrix A. The established conclusions are: if rank(A) = n, then rank(adj(A)) = n; if rank(A) = n-1, then rank(adj(A)) = 1; and if rank(A) < n-1, then rank(adj(A)) = 0. The discussion also emphasizes the use of Jordan Normal Form, where A can be expressed as A = PJP-1, with P being an invertible matrix and J an upper triangular matrix containing eigenvalues on its diagonal.

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Yankel
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Hello all

I need to prove these 3 statements, and I don't know how to start...

A is an nxn matrix:

1) if rank(A)=n then rank(adj(A))=n
2) if rank(A)=n-1 then rank(adj(A))=1
2) if rank(A)<n-1 then rank(adj(A))=0

thanks...:confused:
 
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Yankel said:
Hello all

I need to prove these 3 statements, and I don't know how to start...

A is an nxn matrix:

1) if rank(A)=n then rank(adj(A))=n
2) if rank(A)=n-1 then rank(adj(A))=1
2) if rank(A)<n-1 then rank(adj(A))=0

thanks...:confused:

Write A in Jordan Normal Form:
$$A = P J P^{-1}$$
where P is an invertible matrix and J is an upper triangular matrix with its eigenvalues on its diagonal, and more specifically J consists of Jordan blocks.

If rank(A)=n-1, then J can be written with a row consisting of zeroes, a column consisting of zeroes, and the corresponding minor will be non-zero.

If rank(A)<n-1, then J can be written with at least two rows consisting of zeroes, and at least two columns consisting of zeroes.
 

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