Problem 28 - Differential Equations and Linear Algebra

Question

Given an example of a matrix function of the specified form. (Many examples may be possible.) \(2 \times 3\) matrix function defined only for values of \(t\) with \(-2 \leq t<3\)

Step-by-Step Solution

Answer

\[ \begin{bmatrix} t + 3 & t^2 - 1 & 2t \\ 4 - t & t^3 - 2t & 5t - 2 \end{bmatrix} \]

Step 1: Matrix Function Example

\[ \begin{bmatrix} t + 3 & t^2 - 1 & 2t \\ 4 - t & t^3 - 2t & 5t - 2 \end{bmatrix} \] Now, let's verify that the function is defined for values of \(t\) in the range \(-2 \leq t < 3\). For the given matrix function, \[ \begin{bmatrix} t + 3 & t^2 - 1 & 2t \\ 4 - t & t^3 - 2t & 5t - 2 \end{bmatrix} \] Each element of the matrix is defined for values of \(t\) within the given range and the function is valid for the specified form. Therefore, we have successfully come up with an example of a 2x3 matrix function satisfying the conditions of the exercise.

Step 2: Set up the problem

Write out the given matrices or vectors in standard form.

Step 3: Perform the matrix operations

Execute the required operations, showing key intermediate steps.

Step 4: Analyze the result

Interpret the result in terms of the original problem.

Step 5: State the conclusion

Clearly state the final answer.

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with the study and manipulation of matrices. Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are a fundamental tool in various fields including algebra, geometry, statistics, and engineering.

When working with matrices, certain rules and operations must be followed. For instance, addition and subtraction of matrices are only possible when the matrices have the same dimensions. Similarly, matrix multiplication follows specific rules where the number of columns in the first matrix must match the number of rows in the second matrix.

In the context of our textbook exercise, a 2x3 matrix is a matrix with 2 rows and 3 columns. Matrix algebra allows us to manipulate this matrix with respect to 't', a variable. Each element of the matrix is an expression involving 't', and when 't' changes, so does the matrix. This leads to the concept of a matrix function—a function where the input is a variable that affects each entry of the matrix.

Range of a Function

The range of a function in mathematics refers to the set of all possible output values that a function can produce. This is distinct from the domain, which is the set of all possible input values that the function can accept. For a typical function, the range is determined by evaluating the function across its domain and observing the outputs.

In our example, the function is a matrix function. The domain is specified as the values of 't' such that (-2 leq t < 3). This means our function will only produce outputs (that is, matrices) for these values of 't'. While the matrix function's range isn't numerically defined in a simple form like a single-variable function, it conceptually represents all the possible 2x3 matrices that can be generated by the given polynomial expressions when 't' is substituted within the given domain.

Polynomial Matrix

A polynomial matrix refers to a matrix where each element is a polynomial function of one or more variables. Polynomials are mathematical expressions consisting of variables and coefficients, that involve only the operations of addition, subtraction, multiplication, and positive integer exponents.

In the provided textbook solution, the 2x3 matrix is a polynomial matrix because each element is a polynomial in the variable 't' (e.g., (t^3 - 2t)). Such matrices are particularly interesting in various applications as they combine the properties of polynomials with the structural benefits of matrices. These benefits include being able to perform operations such as addition, multiplication, and even finding roots in some cases, much like we would with single-variable polynomials but within a matrix framework. This polynomial matrix concept allows us to extend our algebraic tools when dealing with systems of equations, dynamical systems, and other scenarios where matrices are used to model complex relationships.