Problem 43 - Differential Equations and Linear Algebra

Question

Determine the derivative of the given matrix function. $$A(t)=\left[\begin{array}{rcc} \sin t & \cos t & 0 \\ -\cos t & \sin t & t \\ 0 & 3 t & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The derivative of the given matrix function is: \(\frac{d}{dt}A(t)=\left[\begin{array}{ccc} \cos t & -\sin t & 0 \\\ \sin t & \cos t & 1 \\\ 0 & 3 & 0 \end{array}\right]\).

Step 1: Find the Derivative for Each Element Individually

To find the derivative for each element individually, we will compute the standard elementary derivatives for each function inside the matrix. Derivative of \(\sin t\) is \(\cos t\), Derivative of \(\cos t\) is \(-\sin t\), Derivative of 0 is 0, Derivative of \(-\cos t\) is \(\sin t\), Derivative of \(t\) is 1, Derivative of 3t is 3.

Step 2: Assemble the Derivatives in the Matrix

Now that we have all the derivatives of the individual elements, let's assemble them in the matrix format. Use the derivatives you found in Step 1 to fill in each position of the matrix. \[ \frac{d}{dt}A(t)=\left[\begin{array}{ccc} \cos t & -\sin t & 0 \\\ \sin t & \cos t & 1 \\\ 0 & 3 & 0 \end{array}\right] \] So the result of the derivative of the given matrix function will be: \[ \frac{d}{dt}A(t)=\left[\begin{array}{ccc} \cos t & -\sin t & 0 \\\ \sin t & \cos t & 1 \\\ 0 & 3 & 0 \end{array}\right] \]

Key Concepts

Derivative of Matrix Function

Understanding the derivative of a matrix function is a fundamental concept in linear algebra and calculus. Just like with scalar functions, the derivative of matrix-valued functions captures its rate of change. However, with matrices, we must look at the rate of change element-wise because a matrix is comprised of multiple scalar functions.

For the given matrix function, we apply the concept of derivative to each entry of the matrix individually. This means for the matrix A(t), each function such as \(\sin t\), \(\cos t\), and \(t\) inside the matrix is differentiated as if it were a standalone scalar function. The resulting derivatives are then placed in the same positions as their respective functions in the original matrix, creating a new matrix which represents the derivative of the original matrix function.

The exercise helps students appreciate the simplicity in working with derivatives in the matrix setting - the operations are akin to what students have already mastered in a scalar context, which provides an intuitive stepping stone to more advanced matrix calculus concepts.

Elementary Derivatives

Elementary derivatives refer to the derivatives of basic functions such as polynomial, exponential, trigonometric, and logarithmic functions. The exercise shows how algebra and trigonometry are extended into matrix calculus. By finding the elementary derivatives of basic trigonometric functions like \(\sin t\) and \(\cos t\), we leverage what students already know from single-variable calculus.

When evaluating the derivative of the matrix function A(t), we take the derivative of each element within the matrix as follows:

• The derivative of \(\sin t\) with respect to time t is \(\cos t\).
• The derivative of \(\cos t\) with respect to t is \(\-\sin t\).
• For constants like 0 and 1, the derivative is simply 0.
• For linear terms like \(t\) and \(3t\), the derivatives are 1 and 3, respectively.

This step-by-step approach reinforces the concept that differentiation of matrices is done entry-wise, mirroring the rules of elementary derivatives applicable to single-variable functions.

Assembling Matrix

Once the derivatives of individual elements are calculated, the next and final step is to assemble these derivatives into a new matrix. This is done by placing each derivative into its corresponding location as per the original matrix structure. In essence, we are creating a matrix of derivatives, which is also referred to as the Jacobian matrix in multivariable calculus when dealing with vectors.

The process of assembling the matrix is methodical and ensures that the structure of the original matrix is preserved. For our example, each derivative taken from Step 1 is methodically placed to mirror the positions of the original matrix A(t). This yields the derivative matrix:

\[\frac{d}{dt}A(t)=\left[\begin{array}{ccc}\cos t & -\sin t & 0 \r\sin t & \cos t & 1 \r0 & 3 & 0\end{array}\right]\]

It is crucial for students to practice this step thoroughly because it builds the foundation for further studies in linear transformations and differential equations where such matrices play a key role. The exercise demonstrates how the derivatives of simpler, well-known functions can be systematically combined to handle more complex structures such as matrices.