- #1
Bertrandkis
- 25
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Question 1
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)
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)