Problem 1 (The Gershgorin circle theorem}. Theorem 1 {Gershgorin circle theorem). Let A E MT, have real or complex entries. Dene the spectrum of A, Sp(A), as the set of all its eigenvalues. For 2' = 1, . _ . ,n, dene the ith deleted row sum of A as n not) = Elna .1'=1 #11 the ith Gershgorin disk of A as T1;(A}={z E 'C : |z (1,43 r,(A)} and the Gershgorin set of A as Then, for all A E MnIC} and for all A E Sp[A), there exists k E {1, . _ _ ,n} such that l)- Ell-kl S Tk(A)1 i.e., A E 1"},(A) and thus A E HA}. Since this is true for all A, we have Sp(A) g rm}. To develop an understanding of the use of this theorem, we will use the two following matrices 1 _1 2 1 i A = (I 2) and B = 1 1 + 22' 0 _ 1 2 1 (3:) Compute all deleted row Sums for A and B. (b) F0r each of A and B, plot the Gershgorin disks. [The plot does not need to be extremely precise, but it needs to be roughly to scale and we need to be able to understand what it is you are plotting] (c) Repeat the plots from above, this time highlighting the Gershgorin sets of A and B. (d) Compute the eigenvalues of A and B (yOu may use numerical software to compute the eigenvalues of B) and repeat the plot above, this time adding the eigenvalues. (e) For matrix B, make tw0 side by side plots as in (d) using B and BT. What do you observe