For each of these questions, please explain how to do it and what is the underlying concept that I need to know to master any questions like that. Thank you

\f8. (a) Suppose that a 3 A = and B = A C b are similar matrices, find the numbers a and b. (b) Suppose that C d O O ON O C = 4 0 and D = O 2 are similar matrices, find the numbers c, d, and e.\f10. Let Use five different tests (the eigenvalue test, the energy test, S = AT A, the determinant test, the pivot test) to show that S is positive definite. -1 2 11. Let S = |[ 0 c B = O 0 (a) Find the smallest integer such that S is positive definite. (b) Find the Cholesky factorization of S when c equals to the result of part a). 12. Let f(z,y,2) = 22 + y% + 422 + c(zy + 2y2). (a) Find the conditions on such that f(z,y,z) > 0 for all z,y, 2 R with 22 + 32 + 22 > 0. (Hint: Write f(z,y,2) = x! Sx for a symmetric matrix S.) (b) Find the conditions on such that f(x,y,z) > 2z for all z,y,z R with z? + y? + 22 0