Let T = -2 1 1 0 2 -3 0 3 Let w = and...

 

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Let T = -2 1 1 0 2 -3 0 3 Let w = " and denote I3 = (c) Express an arbitrary vector v = nation of eigenvectors of T. " (a) i). Calculate det (T-xI3) for any x 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. 6: 1 0 0 0 1 0 01 [ 1] (a) Show that ATWA <=w. (b) Use part (a), or otherwise, show that A-1 = w-¹ ATw. (c) We define u * v = = uTwv for any u, v in R2. Show that for all x, y € R², i). x * y = y* x; and ii). (Ax) * (Ay) = x*y. a B bin R³, where a, b, c € R, as a linear combi- and let A be any 2 × 2 matrix over R with det A = 1. Let T = -2 1 1 0 2 -3 0 3 Let w = " and denote I3 = (c) Express an arbitrary vector v = nation of eigenvectors of T. " (a) i). Calculate det (T-xI3) for any x 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. 6: 1 0 0 0 1 0 01 [ 1] (a) Show that ATWA <=w. (b) Use part (a), or otherwise, show that A-1 = w-¹ ATw. (c) We define u * v = = uTwv for any u, v in R2. Show that for all x, y € R², i). x * y = y* x; and ii). (Ax) * (Ay) = x*y. a B bin R³, where a, b, c € R, as a linear combi- and let A be any 2 × 2 matrix over R with det A = 1.