Problem 72 - Intermediate Algebra

Question

Use a calculator with matrix capabilities. Evaluate each determinant. See Using Your Calculator: Evaluating Determinants. $$ \left|\begin{array}{rrr} 13 & -27 & 62 \\ -38 & 27 & -52 \\ 10 & -300 & 42 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 43144.

Step 1: Matrix Entry

First, carefully enter the given matrix into your calculator. Ensure that each element is accurately input, matching the matrix format below:\[\begin{bmatrix} 13 & -27 & 62 \ -38 & 27 & -52 \ 10 & -300 & 42 \end{bmatrix}\]

Step 2: Access Determinant Function

Locate the determinant function on your calculator. This will often be found in the matrix or math menu, depending on your calculator model.

Step 3: Compute Determinant

Apply the determinant function to the matrix you entered. The calculator will compute the determinant value of the matrix, using its internal numerical methods.

Step 4: Record the Result

Once the calculator displays the determinant, note the value. This is the determinant of the given matrix.

Key Concepts

Matrix Entry

When dealing with matrices in a mathematical context, the first step is to input the matrix values into your calculator accurately. This is known as "matrix entry." A matrix is a rectangular array of numbers arranged in rows and columns. When entering a matrix, pay close attention to each numeral's placement. For example: - The top row: 13, -27, 62 - The middle row: -38, 27, -52 - The bottom row: 10, -300, 42 Each number, known as a matrix entry, needs to be typed in the correct spot. Misplacing just one number can lead to incorrect results. Use the matrix function on the calculator to set the matrix dimensions, in this case, 3 rows by 3 columns. Double-check your numbers once entered to minimize errors.

Calculator Functions

A calculator with matrix capabilities can perform complex mathematical operations easily and accurately. Among these is the ability to compute determinants with the push of a button. Here's how you can access these functions: - **Matrix Menu**: Many calculators have a dedicated matrix or math menu. This is where you often find options to initialize matrices and calculate their determinants. - **Function Navigation**: Use arrow keys or touchscreens to navigate through the menu. This step is crucial, as finding the correct function saves time and ensures correct operations. - **Input Verification**: After selecting the determinant function, confirm your matrix selection. Make sure all inputs are correct and that the calculator is using the matrix you intended. Following these steps allows seamless utilization of calculator functions, making complex calculations faster and more user-friendly.

Evaluating Determinants

To evaluate a matrix determinant means finding a special numerical value that can provide insights into the properties of the matrix. Determinants are particularly important in linear algebra for several reasons:

• They help determine if a matrix is invertible. A non-zero determinant means the matrix can be inverted.
• They reveal important information about the linear transformations represented by the matrix, such as scale changes and rotations.
• In systems of equations, a zero determinant indicates that the system doesn't have a unique solution. Instead, it may have no solution or infinitely many.

To compute it, use your calculator's determinant function, which generally relies on numerical methods designed to handle large numbers and complex calculations efficiently. Once calculated, recording this value can be useful for further mathematical problem-solving and analysis.