Practice 4 Solutions

1 3 1 1 1 n 3 1 1 an 2 find an where a is the matrix

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Unformatted text preview: Ay where 220 A = 1 1 0 , 211 λ = 0, 1, 3 eAt 1 y ( 0) = 1 1 1 −2 0 0 00 440 et e 3t 1 −1 2 0 + 0 0 0 + 2 2 0 = 3 2 6 −1 20 −1 −3 2 550 −1 0 8 t 3t 1 e e 0 + 4 y = 1 + 3 2 6 1 −2 10 [6] Solve the di erential equation y = Ay where 220 A = 1 1 0 , 111 0 y ( 0) = 1 1 λ = 0, 1, 3 eAt 1 −2 0 0 00 440 t 3t e e 1 2 0 + 0 0 0 + 2 2 0 = −1 3 2 6 0 00 −1 −1 2 330 −2 0 4 t 3t 1 + e 0 + e 2 2 y= 3 2 6 0 1 3 [7] Solve the di erential equation y = Ay where 210 A = 0 1 1 , 221 1 y ( 0) = 1 1 , 3 × 3 Exercise Set C (distinct roots), November λ = 0, 1, 3 eAt −1 −1 1 2 0 −1 221 t 3t 1 e e 2 −2 + −2 0 1 + 2 2 1 = 2 3 2 6 −2 −2 2 00 0 442 −1 1 5 t 3t 1 e e −1 + 5 y = 2 + 3 2 6 −2 0 10 [8] Solve the di erential equation y = Ay where 111 A = 0 2 1 , 111 0 y ( 0) = 1 1 λ = 0, 1, 3 eAt 1 0 −1 1 −1 0 132 t 3t e e 1 −1 1 0 + 1 3 2 = 1 0 −1 + 3 2 6 −2 0 2 1 −1 0 132 −1 −1 5 t 3t 1 e e 1 + 5 y = −1 + 3 2 6 2 −1 5 , 3 × 3 Exercise Set D (repeated roots), November 3 × 3 Exercise Set D (repeated roots) Linear Algebra, Dave Bayer, November , [1] Find An where A is the matrix 211 A = 1 1 0 012 222 2 −2 −2 0 0 0 3 1 n 1 1 1 + −1 3 −1 + 1 −1 −1 = 4 4 2 111 −1 −1 3 −1 1 1 n λ = 3, 1, 1 An [2] Find An where A is the matrix 211 A = 0 2 1 212 λ = 4, 1, 1 An 333 6 −3 −3 0 0 0 1 n 4n 2 2 2 + −2 7 −2 + −2 1 1 = 9 9 3 444 −4 −4 5 2 −1 −1 [3] Find An where A is the matrix 201 A = 2 1 1 102 λ = 3, 1, 1 An...
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