Question

Evaluate the determinant of the matrix. Do not use a graphing utility. $$ \left[\begin{array}{rrrr} 2 & 3 & -1 & -1 \\ 0 & -1 & -3 & 5 \\ 0 & 0 & -2 & 7 \\ 0 & 0 & 0 & -4 \end{array}\right] $$

Short answer

The determinant of the given matrix is 16.

Step-by-step solution

Identify the Diagonal Elements

In the given matrix, identify the elements on the diagonal. They are 2, -1, -2, and -4.

Calculate the determinant

The determinant of an upper triangular matrix is the product of the values on its diagonal. So, multiply these four values together. The product of 2, -1, -2, and -4 is \(2 \times -1 \times -2 \times -4 = 16\).

Key concepts

Upper Triangular Matrix

An upper triangular matrix is a type of square matrix where all the entries below the main diagonal are zero. This means that for an upper triangular matrix, only the diagonal and the elements above it may contain non-zero values. Here are some key characteristics of upper triangular matrices:

- They are always square matrices, meaning they have an equal number of rows and columns.
- The simplicity of their structure allows for easier computation of determinants.

In the context of determinating the matrix, knowing it is upper triangular is crucial because it simplifies the determinant calculation. Specifically, the determinant can be found simply by multiplying its diagonal elements. This makes upper triangular matrices powerful tools in linear algebra, saving time and reducing computation complexity.

Diagonal Elements

Diagonal elements of a matrix are those entries that form the line from the top left to the bottom right corner of a square matrix. In mathematical notation, these are the elements \(a_{ii}\) where the row and column indices are equal. For example, in the matrix from the given problem,

- The diagonal elements are 2, -1, -2, and -4.

These diagonal elements have a significant role in determining the determinant of an upper triangular matrix. Since only these elements are involved in the multiplication for finding a determinant, it simplifies the task significantly. Understanding this property helps you appreciate why certain matrices are easier to work with and why the determinant can be calculated the way it is.

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra used to combine matrices and is central to computing things like transformations and determinants. However, in the context of the current exercise, matrix multiplication is used slightly differently.

When calculating the determinant of an upper triangular matrix, what you're really doing is multiplying the diagonal elements together. It's a unique use of multiplication that highlights a special property of triangular matrices. Instead of using standard matrix multiplication rules to find a traditional matrix product, this exercise only requires scalar multiplication of specified elements.

This method underlines an intersection between the operation of matrix multiplication and calculating determinants, emphasizing how certain properties of matrices can aid in computational efficiency.