Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrr} 3 & 0 & 0 \\ -2 & 5 & 0 \\ 12 & 5 & 7 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 105.

Step 1: Identify the type of matrix

Look at the given matrix and determine its type. In this case, it is a upper triangular matrix because all the elements below the diagonal are zero.

Step 2: Calculating the determinant of a triangular matrix

In the case of a triangular matrix, the determinant is the product of the diagonal elements. The diagonal elements of the given matrix are 3, 5 and 7.

Step 3: Multiply the diagonal elements

Multiply the numbers obtained in the previous step. So, \(3 * 5 * 7 = 105\).

Key Concepts

Upper Triangular Matrix

Understanding the structure of different matrices is essential in algebra, particularly when working with determinants. An upper triangular matrix is a type of square matrix where all the elements below the main diagonal are zero. This means that any element a_{ij}, where i is the row index and j is the column index, will be zero if i is greater than j.

For example, consider this matrix:
\[\left|\begin{array}{ccc}1 & 2 & 3 \0 & 4 & 5 \0 & 0 & 6\end{array}\right|\]
This matrix is upper triangular because the elements below the main diagonal (\(1, 4, 6\)) are all zero. Upper triangular matrices are especially important because their determinants can be calculated more easily, which is particularly useful when solving systems of linear equations or finding eigenvalues.

Matrix Determinant Calculation

Matrix determinant calculation is a fundamental concept in linear algebra. The determinant is a scalar value that can be computed from the elements of a square matrix. It gives us important information about the matrix, such as whether it is invertible and the volume factor of the linear transformation represented by the matrix.

There are various methods for calculating a determinant, but one of the simplest is for an upper triangular matrix. The determinant of an upper triangular matrix—or its counterpart, a lower triangular matrix—is simply the product of the elements on the main diagonal. Here's the formula for the determinant of an upper triangular matrix, where a_{ii} represents the elements on the diagonal:
\[\text{det}(A) = a_{11} \cdot a_{22} \cdot ... \cdot a_{nn}\]
This formula significantly simplifies the determinant calculation process, as it avoids the need for complex operations such as row reduction or expansion by minors that are required for other types of matrices.

Graphing Utility in Algebra

In today's technological age, graphing utilities can be invaluable tools in algebra. These utilities, which may come in the form of handheld devices or software programs, allow for the graphical representation of equations and inequalities, as well as for performing complex calculations like matrix operations and finding determinants.

A graphing utility can be particularly helpful when dealing with high-order matrices where manual calculations would be cumbersome. For students learning these concepts, graphing utilities offer a practical way to verify their work and understand the algebraic structures visually. By entering the matrix into the utility and executing a specific function, one can quickly find the determinant, saving time and reducing potential errors.

It is important for students to learn both the theoretical calculation methods and to know how to use modern tools effectively. Mastery of graphing utilities in algebra ensures that students can choose the most efficient method for the problem at hand, understanding the underlying mathematics while benefiting from technology's convenience.