Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rrr} -1 & 4 & 0 \\ 0 & 2 & 3 \\ 0 & 0 & -3 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 6.

Step 1: Understanding the Matrix

The given matrix is a 3x3 upper triangular matrix:\[\begin{array}{rrr}-1 & 4 & 0 \0 & 2 & 3 \0 & 0 & -3 \\end{array}\]

Step 2: Determine the Property of Upper Triangular Matrix

In an upper triangular matrix, the determinant is simply the product of the diagonal elements. This property stems from the fact that all elements below the main diagonal are zeroes.

Step 3: Calculate the Product of Diagonal Elements

Identify the diagonal elements: -1, 2, and -3. Then, multiply these elements to find the determinant:\[-1 \times 2 \times (-3) = 6\]

Key Concepts

Upper Triangular Matrix

An upper triangular matrix is a type of square matrix where all the elements below the main diagonal are zero. Imagine a right triangle with its base coinciding with the bottom edge of the matrix; everything below the hypotenuse or main diagonal is zero. This type of structure allows certain mathematical operations, like finding the determinant, to be much simpler than with a regular matrix.
Examples of upper triangular matrices include:

• 2x2 matrix: \( \begin{pmatrix} a & b \ 0 & c \end{pmatrix} \)
• 3x3 matrix: \( \begin{pmatrix} a & b & c \ 0 & e & f \ 0 & 0 & i \end{pmatrix} \)

This pattern makes them very efficient for solving linear equations and performing matrix factorizations. The simplicity lies in the fact that once you know the diagonal elements of an upper triangular matrix, you have almost all the information you need!

Diagonal Elements

Diagonal elements in a matrix refer to the numbers that sit along the line stretching from the top-left to the bottom-right of the matrix. These elements are key to many properties and operations involving matrices. In particular, these play a crucial role in calculating determinants, especially in triangle matrices (upper or lower).
For example, in the matrix \( \begin{pmatrix} -1 & 4 & 0 \ 0 & 2 & 3 \ 0 & 0 & -3 \end{pmatrix} \), the diagonal elements are -1, 2, and -3.
When calculating the determinant of an upper triangular matrix, you only need to multiply these diagonal elements together. This is because the zeroes below the diagonal mean you don’t need to worry about complex multiplication and addition of other numbers.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a third matrix. This operation follows specific rules but is not typically performed element-by-element as in simple arithmetic multiplication.
The rows of the first matrix are combined with the columns of the second by multiplying corresponding elements and then adding the products to achieve a single number for their position in the resulting matrix. It's important to note that for two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix.
However, when it comes to diagonal elements of a matrix, in particular with upper triangular matrices, their multiplication for calculating the determinant is straightforward. Here you simply multiply the diagonal elements directly without following the typical rules for matrix multiplication, such as was done in the original problem. This unique scenario simplifies what can often be a complex process.