Problem 55 - College Algebra and Calculus: An Applied Approach

Question

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrr} 7 & 0 & -14 \\ -2 & 5 & 4 \\ -6 & 2 & 12 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 0.

Step 1: Arrange the Matrix

To begin with, organize the matrix in the same order as mentioned in the exercise: \( \left[ \begin{array}{ccc} 7 & 0 & -14 \ -2 & 5 & 4 \ -6 & 2 & 12 \end{array} \right] \).

Step 2: Choose Correct Operation

Next, make sure to select the corresponding operation in the graphing utility software or calculator, that is, calculate the determinant of the matrix.

Step 3: Input the Matrix

Now, input the matrix carefully in the calculator or the software. Double check the entries to make sure there's no mistake.

Step 4: Compute the Determinant

Carry out the operation to calculate the determinant of the matrix. The graphing utility or software will produce the result.

Step 5: Check the Result

Verify the answer. Depending on the software or calculator being used, there may be discrepancies due to rounding errors or earlier mistakes in input. If the result is unexpected, consider redoing the computation to correct any possible errors.

Key Concepts

Matrix Determinant Calculation

Calculating the determinant of a matrix is a fundamental procedure in linear algebra, often symbolizing the scale of transformation represented by the matrix. When you encounter a square matrix, such as the one provided in the exercise, evaluating its determinant gives you a scalar value that can have various interpretations.

For a 3x3 matrix like the one we're considering, which looks like \[\left|\begin{array}{ccc} 7 & 0 & -14 \ -2 & 5 & 4 \ -6 & 2 & 12 \end{array}\right|\], the determinant provides insights into whether the system of equations corresponding to the matrix is solvable, and if the matrix can be inverted. The determinant is calculated through a process called 'expansion by minors', among others. Yet, in practice, modern tools like graphing calculators simplify this evaluation significantly, sparing students from manual computation.

Graphing Utility Use

Graphing utilities, including calculators and software, are invaluable tools when confronted with complex mathematical computations, particularly those in linear algebra. To effectively use a graphing utility to evaluate a determinant, it's crucial to familiarize yourself with the device's matrix functions.

Follow these steps: input the matrix exactly as presented, select the determinant calculation function, and execute the command. These utilities often go beyond just calculating determinates, capable of performing a range of operations such as matrix addition, multiplication, inversion, and more. However, when using them, attentiveness is key to avoid input errors, and verifying results is always advised to ensure accuracy in your linear algebra work.

Linear Algebra

Linear algebra is the branch of mathematics concerned with vector spaces, linear mappings, and systems of linear equations. Within this field, the concept of matrices and their determinants is a cornerstone.

Understanding how matrices encode information about linear transformations and how the determinant relates to properties like invertibility and the behavior of eigenvalues is fundamental. The determinant can reveal whether a set of vectors is linearly independent or if they span a vector space. In essence, linear algebra equips us with the tools to tackle a wide array of problems in mathematics, physics, engineering, and computer science, with matrix operations being part of the core skill set one must acquire in this domain.