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6. Read the whole problem before you answer the questions within. Recall that: Given a square matrix T , and a positive integer k = 1,2.., the kh -power of T is the matrix obtained by multiplying T by itself k-times, using matrix multiplication. That is, Tk = T·T.....T k-times, where · stands for matrix multiplication In this problem, we consider the matrices 5 0 D = D-; :) p-1 A = -1 3 P = -1 and Indeed, the matrices A and D are related as follows A = PDP-'. (*) You can use the fact stated in (*) and you DO NOT need to verify it during this test. (a) Use (*) to show that A? = PD²P-l holds. (b) Is the statement Ak = PD*P¯' true for any positive integer k > 3? Explain (briefly). (c) Compute the matrix D² and then find a formula for the matrix D“, for all k > 3. The entries of the matrix Dk should be expressed in terms of the power k. (d) Use parts (b) and (c) to compute A1º. You may leave the entries of A1º as powers of certain numbers, but the output has to be a single 2 x 2 matrix. (e) Find a formula for A' (your output is a 2 × 2 matrix whose entries are to be deter- mined, in terms of k).

6. Read the whole problem before you answer the questions within. Recall that: Given a square matrix T , and a positive integer k = 1,2.., the kh -power of T is the matrix obtained by multiplying T by itself k-times, using matrix multiplication. That is, Tk = T·T.....T k-times, where · stands for matrix multiplication In this problem, we consider the matrices 5 0 D = D-; :) p-1 A = -1 3 P = -1 and Indeed, the matrices A and D are related as follows A = PDP-'. (*) You can use the fact stated in (*) and you DO NOT need to verify it during this test. (a) Use (*) to show that A? = PD²P-l holds. (b) Is the statement Ak = PD*P¯' true for any positive integer k > 3? Explain (briefly). (c) Compute the matrix D² and then find a formula for the matrix D“, for all k > 3. The entries of the matrix Dk should be expressed in terms of the power k. (d) Use parts (b) and (c) to compute A1º. You may leave the entries of A1º as powers of certain numbers, but the output has to be a single 2 x 2 matrix. (e) Find a formula for A' (your output is a 2 × 2 matrix whose entries are to be deter- mined, in terms of k).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 30E
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6. Read the whole problem before you answer the questions within. Recall that: Given a square matrix T , and a positive integer k = 1,2.., the kh -power of T is the matrix obtained by multiplying T by itself k-times, using matrix multiplication. That is, Tk = T·T.....T k-times, where · stands for matrix multiplication In this problem, we consider the matrices 5 0 D = D-; :) p-1 A = -1 3 P = -1 and Indeed, the matrices A and D are related as follows A = PDP-'. (*) You can use the fact stated in (*) and you DO NOT need to verify it during this test. (a) Use (*) to show that A? = PD²P-l holds. (b) Is the statement Ak = PD*P¯' true for any positive integer k > 3? Explain (briefly). (c) Compute the matrix D² and then find a formula for the matrix D“, for all k > 3. The entries of the matrix Dk should be expressed in terms of the power k. (d) Use parts (b) and (c) to compute A1º. You may leave the entries of A1º as powers of certain numbers, but the output has to be a single 2 x 2 matrix. (e) Find a formula for A' (your output is a 2 × 2 matrix whose entries are to be deter- mined, in terms of k).
Transcribed Image Text:6. Read the whole problem before you answer the questions within. Recall that: Given a square matrix T , and a positive integer k = 1,2.., the kh -power of T is the matrix obtained by multiplying T by itself k-times, using matrix multiplication. That is, Tk = T·T.....T k-times, where · stands for matrix multiplication In this problem, we consider the matrices 5 0 D = D-; :) p-1 A = -1 3 P = -1 and Indeed, the matrices A and D are related as follows A = PDP-'. (*) You can use the fact stated in (*) and you DO NOT need to verify it during this test. (a) Use (*) to show that A? = PD²P-l holds. (b) Is the statement Ak = PD*P¯' true for any positive integer k > 3? Explain (briefly). (c) Compute the matrix D² and then find a formula for the matrix D“, for all k > 3. The entries of the matrix Dk should be expressed in terms of the power k. (d) Use parts (b) and (c) to compute A1º. You may leave the entries of A1º as powers of certain numbers, but the output has to be a single 2 x 2 matrix. (e) Find a formula for A' (your output is a 2 × 2 matrix whose entries are to be deter- mined, in terms of k).
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