Prove the following statements and illustrate them with examples of your own choice. Here, ?1, ? ? ? ?n are the (not necessarily distinct) Eigen values of a given n x n matrix A = [ajk].(a) Trace the sum of the main diagonal entries is called the trace of A. It equals the sum of the Eigen values.(b) ??Spectral shift??. A ?? k1 has the Eigen values ?1 ?? k, ??? ?n ?? k and the same Eigen vectors as A.(c) Scalar multiples, powers. kA has the Eigen values k ?1, ??? k?n. Am (m = 1, 2, ???) has the Eigen values ?1m, ??? ?nm. The Eigen vectors are those of A. (d) Spectral mapping theorem, the "polynomial matrix"Has the Eigen values where j = 1, ??? nm and the same Eigen vectors as A.(e) Peron??s theorem, Show that a Leslie matrix L with positive l12, l13, l21, l32, has a positive Eigen value. (This is a special case of the famous Perron-Frobenius theorem in Sec. 20.7, which is difficult to prove in its general form.).

p(A) = kmA + km-1Am-1 +...+kA + kol P(Aj) = km^j + km-1^;m-1 + ko km-1-++kd; +