(a) We proved in class that similar matrices have the same eigenvalues (including multiplicities). Considering the fact

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A We Proved In Class That Similar Matrices Have The Same Eigenvalues Including Multiplicities Considering The Fact 1 (123.35 KiB) Viewed 69 times

(a) We proved in class that similar matrices have the same eigenvalues (including multiplicities). Considering the fact that the rank of a matrix is the number of its non-zero eigenvalues, we can conclude that similar matrices have the same rank. In this question, we will prove this by using another method. (i) Show rank(AB) = rank(B) if A is invertible. (ii) Show rank(AB) = rank(A) if B is invertible. (iii) Show, by using parts (i) and (ii), that if A is similar to B, then rank(A) = rank(B). (b) In the literature, an invertible matrix is also called a nonsingular matrix. Similarly, a matrix that is not invertible is called singular. Suppose A is similar to B. Prove that A is singular iff B is singular. (c) Prove or disprove: If A is similar to B, then Null(A) = Null(B). (d) Prove or disprove: A is similar to RREF(A).