questions d-f 7. Recall the following statement from class:If A and B are similar matrices, then A and B have the same eigenvalues, eachoccurring with the same algebraic and geometric multiplicities.In this problem, we will complete the proof of this statement by showing that if A is an eigen-value of A (and therefore also of B), then the geometric multiplicity of A as an eigenvalueof A must equal the geometric multiplicity of A as an eigenvalue of B.Suppose B = s-1AS, where A, B, and S are all n x n matrices. Show the(a)following two facts:(i) if v is in ker(B), then Sv is in ker(A).(ii) if w is in ker(A), then S-w is in ker(B).Let v1, V2, . .., Vm be a basis of ker(B). Show that the vectors Sv1, Sv2,Svm, which (by part (a)) are in ker(A), are linearly independent.(b)....(Hint: Start by considering a nontrivial relation c1(Sv1)+c2(Sv2)+· ·+cm(Svm) = 0.What happens if you multiply both sides of this relation by S-1?)(c)Show that the vectors Sv1, Sv2, ., Sv m span ker(A).(Hint: Let w be a vector in ker(A). What can you say about S-lw?)(d)dim ker(A)Combine (b) and (c) to show that if A and B are similar matrices, then= dim ker(B).(e)Show that if A and B are similar, then the matrices A – XIn and B – XInare similar as well.(Hint: Take another look at the proof given in class that similar matrices have thesame characteristic polynomial.)(f)of A (and therefore also of B), then the geometric multiplicity of A as an eigenvalue ofA must equal the geometric multiplicity of as an eigenvalue of B.Use your answers to the previous parts to conclude that if is an eigenvalue

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(d) dim ker(A) = dim ker(B). Combine (b) and (c) to show that if A and B are similar matrices, then 2 (e) are similar as well. Show that if A and B are similar, then the matrices A – XIn and B – In (Hint: Take another look at the proof given in class that similar matrices have the same characteristic polynomial.) (f) of A (and therefore also of B), then the geometric multiplicity of A as an eigenvalue of A must equal the geometric multiplicity of A as an eigenvalue of B. Use your answers to the previous parts to conclude that if A is an eigenvalue