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If A and B are similar matrices, then prove that A is nonsingular if and only if B is nonsingular.

MY SOLUTION:

B is nonsingular if it's columns are linearly independent

P[tex]^{-1}[/tex]AP=B where the main diagonal of B is made of eigenvectors of A.

How does this affect the nonsingularity of A ????? this is where I am stuck.

Question 2

If A and B are similar matrices, Show that if B=PAP[tex]^{-1}[/tex] then det(B)=det(A)

MY SOLUTION

As B=PAP[tex]^{-1}[/tex] this implies that BP=PA

Taking the determinant of both sides.

det(BP)=det(PA)

when expanded, we have det(B).det(P)=det(P)det(A) therefore det(B)=det(A)