Chapter 13: Chapter 13

Problem 301

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Given $\mathrm{A}(\mathrm{t})=\begin{array}{rr}\mid \mathrm{t}^{2} & \cos \mathrm{t} \mid \\ \mid \mathrm{e}^{\mathrm{t}} & \sin \mathrm{t} \mid\end{array}$ Find $(\mathrm{d} \mathrm{A} / \mathrm{dt})$

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The short answer to the question is: $\frac{dA}{dt} = \begin{pmatrix} 2t & -\sin(t) \\ e^t & \cos(t) \end{pmatrix}$

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Recall the formula for finding the derivative of a matrix

To find the derivative of a matrix, we simply need to find the derivative of each element of the matrix with respect to the variable (in this case, t) and place the derivatives in the resulting matrix in their corresponding positions.

Step 2: Find the derivatives of the matrix elements

We will find the derivatives of each element in the matrix A(t): 1. Derivative of $t^2$: using basic power rule: $\frac{d}{dt}(t^2) = 2t$ 2. Derivative of $\cos(t)$: using basic differentiation rules: $\frac{d}{dt}(\cos(t)) = -\sin(t)$ 3. Derivative of $e^t$: using basic exponential function differentiation: $\frac{d}{dt}(e^t) = e^t$ 4. Derivative of $\sin(t)$: using basic differentiation rules: $\frac{d}{dt}(\sin(t)) = \cos(t)$

Step 3: Form the resulting matrix

Now that we have found the derivatives of each element of A(t), we can form the resulting matrix (dA/dt): \[ \frac{dA}{dt} = \begin{pmatrix} 2t & -\sin(t) \\ e^t & \cos(t) \end{pmatrix} \] That's it! The derivative of the given matrix A(t) is: \[ \frac{dA}{dt} = \begin{pmatrix} 2t & -\sin(t) \\ e^t & \cos(t) \end{pmatrix} \]

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Chapter 13: Chapter 13

Given $\mathrm{A}(\mathrm{t})=\begin{array}{rr}\mid \mathrm{t}^{2} & \cos \mathrm{t} \mid \\ \mid \mathrm{e}^{\mathrm{t}} & \sin \mathrm{t} \mid\end{array}$ Find \((\mathrm{d} \mathrm{A} / \mathrm{dt})\)

Short Answer

Step by step solution

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Step 1: Recall the formula for finding the derivative of a matrix

Step 2: Find the derivatives of the matrix elements

Step 3: Form the resulting matrix

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More chapters from the book ‘Linear Algebra’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

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