Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, use the determinant function on a graphing utility. $$ \left|\begin{array}{llll} 1 & 0 & 8 & 9 \\ 0 & 2 & 1 & 0 \\ 1 & 0 & 3 & 0 \\ 0 & 2 & 4 & 3 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given matrix is -84.

Step 1: Understand the Matrix

The given matrix is a 4x4 matrix which looks like this: \[\begin{bmatrix}1 & 0 & 8 & 9 \0 & 2 & 1 & 0 \1 & 0 & 3 & 0 \0 & 2 & 4 & 3\end{bmatrix}\]We are tasked with finding the determinant of this matrix using a graphing utility.

Step 2: Input the Matrix into the Graphing Utility

Enter the elements of the matrix into the graphing utility. Make sure each entry corresponds to the correct position in the matrix. Re-check to ensure no numbers are input incorrectly.

Step 3: Use the Determinant Function

Locate the determinant function within the calculator or graphing utility's menu. Select this function and apply it to the matrix you have inputted. The utility will calculate the determinant for you based on the matrix.

Step 4: Interpret the Result

The graphing utility will provide the determinant of the matrix. Make note of this value as it represents the answer to our problem. Confirm that the calculation is correct by reviewing the steps if necessary.

Key Concepts

4x4 Matrix

When we talk about a 4x4 matrix, we're referring to a square matrix that has four rows and four columns. Each element in this matrix is arranged in a rectangular grid. You might see something like this:

• The first row could be represented as the elements across the horizontal top, like [1, 0, 8, 9].
• The first column might look like the vertical line down, such as [1, 0, 1, 0].

Each element is positioned at the intersection of a specific row and column, making it easier to organize and perform calculations. Having a firm grasp on identifying and writing matrices is fundamental in higher mathematics as it helps in solving complex problems involving systems of equations and transformations in space.
Understanding matrices is crucial when working with linear transformations and various mathematical applications, like computer graphics, physics, and statistics. It's important to know how to manipulate matrices, as this forms the basis for understanding many mathematical operations.

Matrix Determinant Calculation

Calculating the determinant of a matrix is a method to find a specific scalar value that provides important properties about the matrix. Specifically, for a 4x4 matrix, it tells us if the matrix is invertible, which is crucial in solving systems of linear equations or performing transformations. The determinant calculation process involves a combination of multiplications and additions of its elements across specific diagonals while following certain rules.

• Methods such as cofactor expansion can be used for smaller matrices, but a 4x4 matrix can be quite complex to solve by hand.
• Instead, we often rely on calculators or programming tools to carry out these exhaustive calculations efficiently.

For manual calculations, the process can be lengthy and prone to error, thereby using a calculator or software tool can significantly simplify the task, providing reliable and quick results.
When the determinant value is zero, it suggests that the matrix does not have an inverse and the transformation it represents is non-invertible or "singular." On the other hand, a non-zero determinant indicates that the matrix is non-singular and invertible, amongst other things, such as being used to determine volume scaling in transformations.

Graphing Utility

A graphing utility is a powerful tool that can help simplify complex mathematical operations, like determining the determinant of a 4x4 matrix. These utilities can be found in many high-end calculators and computer software designed for mathematical computations.

• These tools allow you to input matrix values exactly as they appear, ensuring that the structure and entries of the matrix are correctly represented.
• Once the values are in place, the utility uses built-in algorithms to calculate the determinant with ease, often in just a few seconds.

To use these utilities effectively, familiarity with their interface is necessary. They often have a menu where specific functions like the determinant can be selected.
Upon inputting the matrix and running the determinant function, the utility outputs a precise result that helps to solve many algebraic problems quickly. This not only aids in verifying manual calculations but also enhances learning by allowing students to focus on understanding concepts rather than being bogged down by lengthy calculations. These utilities are an invaluable resource in both academic and professional settings.