Question

Given a square matrix $A$, $e^{At}$ is defined to be a matrix solution to the differential equation $\frac{d}{dt}e^{At} = Ae^{At}$ with initial condition that at $t = 0$, $e^{At}$ is the identity matrix of the same size as $A$. This exponential of $A$ is an effective way to write the general solution of the differential equation $y' = Ay$, because the solution is given by $y(t) = e^{At}y(0)$. Find $e^{At}$ for $A = \begin{bmatrix} 5 & 1\\ -2 & 3 \end{bmatrix}$, and enter your answer into the variable expAt. Hint: the columns of the matrix satisfy the same differential equation.

          Given a square matrix $A$, $e^{At}$ is defined to be a matrix solution to the differential equation $\frac{d}{dt}e^{At} = Ae^{At}$ with initial condition that at $t = 0$, $e^{At}$ is the identity matrix of the same size as $A$. This exponential of $A$ is an effective way to write the general solution of the differential equation $y' = Ay$, because the solution is given by $y(t) = e^{At}y(0)$. 
Find $e^{At}$ for $A = \begin{bmatrix} 5 & 1\\ -2 & 3 \end{bmatrix}$, and enter your answer into the variable expAt.
Hint: the columns of the matrix satisfy the same differential equation.

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Given a square matrix A, e^At is defined to be a matrix solution to the differential equation (d)/(dt)e^At = Ae^At with initial condition that at t = 0, e^At is the identity matrix of the same size as A. This exponential of A is an effective way to write the general solution of the differential equation y' = Ay, because the solution is given by y(t) = e^Aty(0).
Find e^At for A =
< b m a t r i x >, and enter your answer into the variable expAt.
Hint: the columns of the matrix satisfy the same differential equation.

Added by Jose M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

12/12/2022

Step 1

The characteristic polynomial of A is: $$ \begin{aligned} \det(A - \lambda I) &= \det \begin{bmatrix} 5 - \lambda & 1 \\ -2 & 3 - \lambda \end{bmatrix} \\ &= (5 - \lambda)(3 - \lambda) - (-2)(1) \\ &= \lambda^2 - 8\lambda + 17 \end{aligned} $$ The eigenvalues are Show more…

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Given a square matrix A, $e^{At}$ is defined to be a matrix solution to the differential equation $\frac{d}{dt}e^{At} = Ae^{At}$ with initial condition that at t = 0, $e^{At}$ is the identity matrix of the same size as A. This exponential of A is an effective way to write the general solution of the differential equation y' = Ay, because the solution is given by y(t) = $e^{At}$y(0). Find $e^{At}$ for A = $\begin{bmatrix} 5 & 1 \\ -2 & 3 \end{bmatrix}$ and enter your answer into the variable expAt. Hint: the columns of the matrix satisfy the same differential equation.

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