Question

x n matrices, then A is similar to B if and only if Problem 15 [10 pts] If A, B are two pa(t) = Pb(t).

Answer 1

Problem is Geiren A and Bare two nxn matriceli case A and Bare similar 1 attress by definition of similar of similar matuces que hare S-AS-B s i an invertible mattix. where Now , Palt and Polt) denote the characteristic polynomiall of A and B, respectively. we are then

hect_det 18-+1) det (SLAS- EI) - det (S='CAS - Es)) - dut (S-LA- til) det (5-1) det (A-t I det(s) =(dit si det (A-t I det(s) (By the multiplicative properties of detuminant) 2 Adet (A -EI) dets) (det(sit zdetta-tI) Palt) (sešice debuminante ose feidt number, cancel out) Thoug have obtained balt) = Pelt) etus can be As when Palt)= kg Lt). consider two matrice 6= [. Both have same characteristé polynomial Palt) = Polt) = (1-t2 [ii] → But ON=1 has eigenvalue og Eiginrector v= [:] As can be seen, number of eiginrectors (say, 1) i not equal the number of columns (say, 2) and only one

un B. So, matrix B is not diagonalizable Therefore, there ui no invertible manis s exists such that s-'ASEB Hence, А and Bare not similar