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Department of Mathematics & Statistics University of North Florida Prof. Champak D. Panchal March 14, 2005 MAS 3105, TEST II, NAME ________________ , _________________ 1 For 2 1 3 A 1 2 0 find the inverse matrix and using this inverse solve 3 2 5 2 x1 x 2 3x3 10 x1 2 x 2 20 3x1 2 x 2 5 x3 60 1 1 2 2.Find LU decomposition of A 2 4 1 , write elementary matrices that you use 5 1 15 in this work, and also write down the inverses of these elementary matrices. 3.(i) Find a, b, and c if 2 1 1 o a1 b 2 c 0 o Are three vectors on L.. H . S . linearly indepedent ? 3 21 5 o 1 1 2 0 (i ) Find a, b, c if a 2 b 4 c 7 0 5 1 5 0 Are three vectors on L.. H . S . linearly indepedent ? 4. (i) 2x 3y x If T x 2 y , justify that T ( x y ) T x T y, where y 2 y 5x x and y are vectors in R 2 . (ii) Find the matrix that represents the linear transformation in (i) above. Department of Mathematics & Statistics University of North Florida Prof. Champak D. Panchal March 14, 2005 5.(i) The matrices A and Q are of size n by n. Q is invertible, then we have proved that Rank(A) < Rank(AQ). Using this fact, prove Rank(A) = Rank(AB), where B is an n by n invertible matrix. (ii)For B in (i) above prove that ( B T ) 1 ( B 1 ) T u belong to Span u, v, w? (iii ) Justify that Span u, v, z Span u, v iff z is in Span u, v (iv) (i ) Does (v) A is a 20 by 30 matrix, fill in the following blanks: (a) A maps R into R . (b) -------- < Rank(A) < --------(c) Rank(A) ------- Rank(A ).