Suppose that the 2 x 2 matrix A has the following eigenvalues Ak and eigenvectors Tk:...

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Suppose that the 2 x 2 matrix A has the following eigenvalues Ak and eigenvectors Tk: A₁ = -2₁ ñ₁ = ( ³ ); 3 =(-²). A2 = 1, T2 = (a) Write down the general solution for (r(t), y(t)) of the system I () = A (₂) (b) Sketch the phase plane of this system. Include in your sketch the trajec- tory that passes through the point (0,2). (c) Sketch the graphs of r(t) and y(t) versus t for the solution of this system that satisfies the initial condition z(0) = 0, y(0) = 2. Suppose that the 2 x 2 matrix A has the following eigenvalues Ak and eigenvectors Tk: A₁ = -2₁ ñ₁ = ( ³ ); 3 =(-²). A2 = 1, T2 = (a) Write down the general solution for (r(t), y(t)) of the system I () = A (₂) (b) Sketch the phase plane of this system. Include in your sketch the trajec- tory that passes through the point (0,2). (c) Sketch the graphs of r(t) and y(t) versus t for the solution of this system that satisfies the initial condition z(0) = 0, y(0) = 2.

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