please do 6-11

A is a real 3 x 3 matrix and b E R3. The RREF of the augmented matrix associated with the system 1 0 1 0 Ax=bis D 1 1 1 .Solvethesystem. D 0 0 0 Let A be the matrix given in problem 1. a. Find NulA. b. Find ColAT. 4 a 1) Suppose A = ( a 5 c) where a, b, c E R. Suppose um E R3 are nonzero vectors so that b c 8 Au = 2u and Au = 312. a. Give all of the eigenvalues of A, along with their algebraic and geometric multiplicities. b. detA = Orthogonally diagonalizeA = (3 1). 1 3 2 1 2 Determine whether the matrix A = 1 0 0 is diagonalizable. If it is, then give a nonsingular 0 1 0 matrix P and a diagonal matrix D so that A = PDP'l. a a 1 Find values of a and b so that the matrix % (b 1 b) is orthogonal. 1 a 11 Give the real symmetric matrix A associated with the quadratic form Q: R3 > R given by Q(x) = 2x5 29:19:; + 2x1x3 + 2x3 2x2x3 + 225% Two of the eigenvalues of the matrix A in problem 7 are 4 and 1. Find El and E4 and determine if there is an eigenvalue of A other than 1 and 4. Lewmieicnm@- a. Find projvu. b. Find the angle between V and It (you can give your result in terms of an inverse cosine). c. Find the distance from u to V. d. Find VJ: 1 1 3 1 1 7 10. Let W = span 1 , 3 , 1 1 1 3 a. dimW = b. Find an orthogonal basis for W. 1022 11.U=0122. 2 1 1 4 a. Find the RREF of U. b. rank U = c. dim N111 U =