2. Let T = 1 0 2-3 0 3 and denote I3 = 0 1 0 001 (c) Express an arbitrary vector v = nation of eigenvectors of T. . (a) i). Calculate det (T-213) for any a to find the characteristic polynomial of T; ii). Determine the eigenvalues of T. (b) Find an eigenvector of T associated with each of the eigenvalues of T. a b in R³, where a, b, c ER, as a linear combi-

(b) Find a vector for X;. complete orthonormal set of eigenvectors {v1, V2, V3}, where v; is an eigen-

Help with d please

Let q = - 2xy - y² + 2xz+2yz + z² be a quadratic form on R³ viewed as a polynomial in 3 variables. Find a linear change of variables tou, v, w that puts q into the canonical form in `Sylvester's law of inertia'!. What are the values of the associated indices s, t? Select one: O We let u = √3(x − y + 2), v = x+y, w = (x - y)/2 to find that q = u²+². Hence s = 2, t = 0 in Sylvester's law of intertia. O We let u = x - 2y, v = z+y, w = √3y to find that q=u²v² w². Hence s = 1,t = 2 in Sylvester's law of intertia. O we let u = x, v= √2(x+y), w = x+y+z to find that q = u²v² + w²2. Hence s = 2, t = 1 in Sylvester's law of intertia. O None of the others apply O The quadratic form does not obey the condition to be diagonalisable over R. This is because the minimal polynomial of the corresponding matrix is not a product of distinct linear factors. Hence Sylvester's law of inertia does not apply. By convention, we set s = t = ∞ when this happens.

What are the eigenvectors?  I know the eigenvalues are repeated roots of 1.

Please don't provide handwritten solution ...

Help with part a please

Q6. Find eigenvalues and associated eigenvectors for each matrix. a) A 1 = ² = 11 -1 4-11 0 b) A = 0 1 03-2

Let A = a. Find the characteristic polynomial. b. Find the eigenvalues. c. Find the eigenvectors. d. Find an invertible matrix, P, such that P-'AP is diagonal.

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