03aut-m2sols

# 03aut-m2sols - Question 1 of 7 Page 1 of 7 1(a Let A= 7 4 5...

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Question 1 of 7, Page 1 of 7 Solutions 1. (a) Let A = p 7 - 4 5 - 3 P . Is A invertible? If so, Fnd A - 1 . If not, why not? Solution: det( A ) = 7( - 3) - 5( - 4) = - 1 n = 0. Hence A is invertible. A - 1 = p 3 - 4 5 - 7 P . (b) Let B be the matrix B = 1 0 p 0 1 0 3 0 p . ±or what values of p in R is this matrix invertible? Solution: det( B ) = 1 v v v v 1 0 0 p v v v v + p v v v v 0 1 3 0 v v v v = p - 3 p = - 2 p. Since B is invertible whenver det( B ) n = 0, B is invertible for p n = 0.

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Question 2 of 7, Page 2 of 7 Solutions 2. (a) For each of the following transformations, determine whether T IS a linear transformation. For this part only, you do not need to show your work; simply circle “YES” (if it is a linear transformation) or “NO” (if it is not). 1 . T p x y P = xy YES NO 2 . T x y z = p | x | 0 P YES NO 3 . T p x y P = x + y x + 2 y - 3 x YES NO 4 . T p x y P = 2 y x + 1 x + y YES NO Solution: Only transformation #3 is linear. (b) For each of the transformations you identi±ed in part (a) as a linear trans- formation, determine the associated matrix (with respect to the standard basis). Solution: For transformation #3: T p 1 0 P = 1 1 - 3 , T p 0 1 P = 1 2 0 . Hence the matrix of T with respect to the standard basis is [ T ] = 1 1 1 2 - 3 0 .
Question 3 of 7, Page 3 of 7 Solutions 3. Let T be a linear transformation from R 3 to R 3 so that T 1 0 0 = 2 0 0 , T 1 1 0 = 3 3 0 , T 1 1 1 = 4 4 4 .

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03aut-m2sols - Question 1 of 7 Page 1 of 7 1(a Let A= 7 4 5...

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