Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{rrr} 1 & 0 & 0 \\ -4 & -1 & 0 \\ 5 & 1 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is -5

Step 1: Identify matrix type

Identify the type of the matrix to determine the appropriate method to find the determinant. Here the given Matrix is a 3x3 matrix and it is an upper triangular matrix (All elements below the main diagonal are zero).

Step 2: Apply determinant rule

Apply the rule for finding the determinant of an upper triangular matrix. The determinant of a triangular (upper or lower) matrix can be obtained by simply multiplying the elements of its main diagonal.

Step 3: Calculate determinant

Multiply the elements (1, -1, 5) of the main diagonal of the Matrix. These elements are located at positions (1, 1), (2, 2) and (3, 3) of the matrix. Thus, the determinant is \(1 \times -1 \times 5 = -5\).

Key Concepts

Upper Triangular Matrix

When learning about matrices, one category you'll come across is the upper triangular matrix. These are a type of square matrix, where all elements below the principal diagonal are zero. Visually, imagine a square grid where any numbers that fall on the bottom left triangle of the matrix are zeros, while the upper right triangle, including the diagonal, can contain non-zero entries.

One of the key benefits of an upper triangular matrix is simplicity in calculation, particularly when it comes to finding the determinant. Because of their structure, many operations which may be complex for generic matrices become much more straightforward. So not only do they make your math homework easier, but they're also widely used in numerical analysis and linear algebra - especially when dealing with linear equations and matrix factorization.

Calculating Determinants

The determinant of a matrix is a unique number that provides a lot of information about the characteristics of the matrix. It is particularly useful in linear algebra for analyzing vector spaces and solving systems of linear equations.

For a square matrix, the determinant reflects the volume scaling factor of the linear transformation described by the matrix and can help determine whether a matrix has an inverse. When approaching the calculation of determinants, there are specific rules and techniques based on the structure of the matrix in question. For example, the determinants of diagonal and triangular matrices can be calculated simply by multiplying the elements along the principal diagonal, which is an immediate advantage when dealing with an upper triangular matrix. This method greatly simplifies the computational process, avoiding the more complex expansion by minors or row-reduction algorithms required for other matrix types.

3x3 Matrix

A 3x3 matrix is a matrix with three rows and three columns. Determinants of a 3x3 matrix can often be found using the rule of Sarrus or co-factor expansion but these methods can be unnecessarily complicated when the matrix has a special structure, such as being an upper triangular matrix.

It's also worth noting that the 3x3 matrix is the smallest matrix size from which a meaningful volume analogy can be drawn when thinking about determinants. Thus, when matrices take this form, we have the chance to visualize the concepts we are working with a bit more concretely.

Practical Application

Imagine using a set of 3 vectors to describe a cube in 3D space. The determinant of the matrix comprising these vectors would give us information about the volume of the cube. But remember, with an upper triangular 3x3 matrix, calculating this hypothetical volume is as simple as multiplying the numbers on the diagonal.