Question

(a) Use the fact that det(B) =det(BT) for all matrices B to prove that if λ is an eigenvalue of A, then λ is also an eigenvalue of AT. (You may also use the fact that I-I.) (b) Prove that if the sum of the entries in each row of A = 1, then l is an eigenvalue of A. (Hint: What is A?) (c) Use parts a and b to prove that if the sum of the entries in each column of A is 1 then 1 is an eigenvalue of A

Answer 1

Suppose A has size n, then P_A(x) = det (A - lambda I_n) (characteristic polynomial of A) = det ((A - lambda I_n)^T)(because det (b) = det (B^T) = det (A^T - lambda I^T_n) (I_n^T = I) = det (A^T - lambda I) = P_AT (lambda) (characteristic polynomial of A^T) So A and A^T have same characteristic polynomial {result: suppose A is square matrix . then lambda is an eigen value of A iff P_A (lambda) = 0 } and also note that if we equate P_A(lambda) and P_A^T(lambda) to zero we get the same roots (characteristic roots) and they are nothing but actually eigen values and since the characteristic polynomials are equal, therefore we co9nclude that A and A^T has same eigen values i.e if lambda is eigen value of A, then it is alos an eigen value of A^T let A be the given matrix where sum of entries of earn row