Practice 4 Solutions

Practice 4 Solutions - 2 2 Exercise Set A(distinct roots...

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2 × 2 Exercise Set A (distinct roots), November ±², ±³´µ 2 × 2 Exercise Set A (distinct roots) Linear Algebra, Dave Bayer, November ±², ±³´µ [ 1 ] Find A n where A is the matrix A = ± 1 - 1 3 - 3 ² λ = - 2, 0 A n = (- 2 ) n 2 ± - 1 1 - 3 3 ² + 0 n 2 ± 3 - 1 3 - 1 ² [ 2 ] Find A n where A is the matrix A = ± 1 - 1 - 3 - 1 ² λ = - 2, 2 A n = (- 2 ) n 4 ± 1 1 3 3 ² + 2 n 4 ± 3 - 1 - 3 1 ² [ 3 ] Find A n where A is the matrix A = ± 0 1 2 1 ² λ = - 1, 2 A n = (- 1 ) n 3 ± 2 - 1 - 2 1 ² + 2 n 3 ± 1 1 2 2 ² [ 4 ] Find A n where A is the matrix A = ± 1 2 3 0 ² λ = - 2, 3 A n = (- 2 ) n 5 ± 2 - 2 - 3 3 ² + 3 n 5 ± 3 2 3 2 ² [ 5 ] Find A n where A is the matrix A = ± 2 2 - 2 - 3 ² λ = - 2, 1 A n = (- 2 ) n 3 ± - 1 - 2 2 4 ² + 1 3 ± 4 2 - 2 - 1 ² [ 6 ] Find A n where A is the matrix A = ± - 3 1 - 2 0 ²
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2 × 2 Exercise Set A (distinct roots), November ±², ±³´µ λ = - 2, - 1 A n = (- 2 ) n ± 2 - 1 2 - 1 ² + (- 1 ) n ± - 1 1 - 2 2 ² [ 7 ] Find A n where A is the matrix A = ± 1 1 2 0 ² λ = - 1, 2 A n = (- 1 ) n 3 ± 1 - 1 - 2 2 ² + 2 n 3 ± 2 1 2 1 ² [ 8 ] Find A n where A is the matrix A = ± 0 1 - 2 - 3 ² λ = - 2, - 1 A n = (- 2 ) n ± - 1 - 1 2 2 ² + (- 1 ) n ± 2 1 - 2 - 1 ²
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2 × 2 Exercise Set B (distinct roots), November ±², ±³´µ 2 × 2 Exercise Set B (distinct roots) Linear Algebra, Dave Bayer, November ±², ±³´µ [ 1 ] Find e At where A is the matrix A = ± 3 1 - 3 - 1 ² λ = 0, 2 e At = 1 2 ± - 1 - 1 3 3 ² + e 2 t 2 ± 3 1 - 3 - 1 ² [ 2 ] Find e At where A is the matrix A = ± - 3 2 - 3 2 ² λ = - 1, 0 e At = e - t ± 3 - 2 3 - 2 ² + ± - 2 2 - 3 3 ² [ 3 ] Find e At where A is the matrix A = ± - 1 - 3 - 2 0 ² λ = - 3, 2 e At = e - 3 t 5 ± 3 3 2 2 ² + e 2 t 5 ± 2 - 3 - 2 3 ² [ 4 ] Find e At where A is the matrix A = ± 3 3 - 1 - 1 ² λ = 0, 2 e At = 1 2 ± - 1 - 3 1 3 ² + e 2 t 2 ± 3 3 - 1 - 1 ² [ 5 ] Find e At where A is the matrix A = ± 1 - 1 - 3 - 1 ² λ = - 2, 2 e At = e - 2 t 4 ± 1 1 3 3 ² + e 2 t 4 ± 3 - 1 - 3 1 ² [ 6 ] Find e At where A is the matrix A = ± - 2 2 - 3 3 ²
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2 × 2 Exercise Set B (distinct roots), November ±², ±³´µ λ = 0, 1 e At = ± 3 - 2 3 - 2 ² + e t ± - 2 2 - 3 3 ² [ 7 ] Find e At where A is the matrix A = ± 2 - 2 3 - 3 ² λ = - 1, 0 e At = e - t ± - 2 2 - 3 3 ² + ± 3 - 2 3 - 2 ² [ 8 ] Find e At where A is the matrix A = ± 0 - 1 - 3 2 ² λ = - 1, 3 e At = e - t 4 ± 3 1 3 1 ² + e 3 t 4 ± 1 - 1 - 3 3 ²
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2 × 2 Exercise Set C (distinct roots), November ±², ±³´µ 2 × 2 Exercise Set C (distinct roots) Linear Algebra, Dave Bayer, November ±², ±³´µ [ 1 ] Solve the di¶erential equation y 0 = Ay where A = ± 1 1 2 2 ² , y ( 0 ) = ± 1 0 ² λ = 0, 3 e At = 1 3 ± 2 - 1 - 2 1 ² + e 3 t 3 ± 1 1 2 2 ² y = 1 3 ± 2 - 2 ² + e 3 t 3 ± 1 2 ² [ 2 ] Solve the di¶erential equation y 0 = Ay where A = ± 0 - 1 2 - 3 ² , y ( 0 ) = ± 1 2 ² λ = - 2, - 1 e At = e - 2 t ± - 1 1 - 2 2 ² + e - t ± 2 - 1 2 - 1 ² y = e - 2 t ± 1 2 ² + e - t ± 0 0 ² [ 3 ] Solve the di¶erential equation y 0 = Ay where A = ± 1 - 1 - 3 - 1 ² , y ( 0 ) = ± 1 1 ² λ = - 2, 2 e At = e - 2 t 4 ± 1 1 3 3 ² + e 2 t 4 ± 3 - 1 - 3 1 ² y = e - 2 t 4 ± 2 6 ² + e 2 t 4 ± 2 - 2 ² [ 4 ] Solve the di¶erential equation y 0 = Ay where A = ± - 1 1 2 - 2 ² , y ( 0 ) = ± 1 0 ² λ = - 3, 0 e At = e - 3 t 3 ± 1 - 1 - 2 2 ² + 1 3 ± 2 1 2 1 ² y = e - 3 t 3 ± 1 - 2 ² + 1 3 ± 2 2 ² [ 5 ] Solve the di¶erential equation y 0 = Ay where A = ± 1 1 2 0 ² , y ( 0 ) = ± 1 2 ² λ = - 1, 2 e At = e - t 3 ± 1 - 1 - 2 2 ² + e 2 t 3 ± 2 1 2 1 ² y = e - t 3 ± - 1 2 ² + e 2 t 3 ± 4 4 ²
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2 × 2 Exercise Set C (distinct roots), November ±², ±³´µ [ 6 ]
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Practice 4 Solutions - 2 2 Exercise Set A(distinct roots...

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