Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, use the determinant function on a graphing utility. $$ \left|\begin{array}{rrrr} \frac{1}{2} & 1 & 7 & 4 \\ 0 & \frac{1}{2} & 100 & 5 \\ 0 & 0 & 2 & 2,000 \\ 0 & 0 & 0 & 2 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is \( \frac{1}{2} \).

Step 1: Understanding the Matrix

Firstly, recognize that the given matrix is a triangular matrix, specifically an upper triangular matrix. All the entries below the main diagonal are zeros.

Step 2: Identifying the Determinant Property

Remember that the determinant of a triangular matrix (either upper or lower) is the product of its diagonal elements. This property simplifies the calculation greatly.

Step 3: Extracting Diagonal Elements

List the diagonal elements from the given matrix: \( \frac{1}{2}, \frac{1}{2}, 2, \) and \( 2 \).

Step 4: Calculating the Determinant

Multiply the diagonal elements together to find the determinant: \( \frac{1}{2} \times \frac{1}{2} \times 2 \times 2 = \frac{1}{2} \).

Step 5: Conclusion

After calculating the product of the diagonal elements, we determined that the determinant of the matrix is \( \frac{1}{2} \).

Key Concepts

Triangular Matrix

Understanding the structure of a triangular matrix is key to mastering determinant calculations. A triangular matrix is a special type of square matrix where all the elements above or below its main diagonal are zero. There are two types of triangular matrices: upper triangular and lower triangular. In our case, we have an upper triangular matrix because all elements below the diagonal are zeros. This characteristic makes such matrices easier to work with, particularly when it comes to finding determinants. When a matrix is triangular, either upper or lower, the task of calculating its determinant becomes straightforward.

Diagonal Elements

Diagonal elements are those that lie on the main diagonal of a matrix. This diagonal runs from the top-left to the bottom-right corner of a square matrix. For our matrix, the diagonal elements are crucial for determining the determinant. They are:

• \(\frac{1}{2}\)
• \(\frac{1}{2}\)
• 2
• 2

These specific elements are the focus because, in triangular matrices, the determinant is simply the product of these diagonal elements. This simplification is one of the benefits of working with triangular matrices, making the calculation of determinants more efficient and less error-prone.

Matrix Properties

Matrices possess several interesting properties, especially when they are categorized into types like triangular matrices. Apart from the simplification of determinant calculation, triangular matrices have other useful properties:

• Inverse: If all diagonal elements are non-zero, the matrix is invertible.
• Rank: The rank of a triangular matrix is equal to the number of its non-zero diagonal elements.
• Eigenvalues: For triangular matrices, the eigenvalues are directly the diagonal elements.

These properties make triangular matrices highly beneficial in linear algebra applications, particularly in solving linear equations, finding eigenvalues, and simplifying computations. Understanding these properties and how they interact aids in a deeper comprehension of matrix operations and their implications in advanced mathematical problems.