Problem 8 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 0.2 & 0.4 \\ -0.4 & -0.8 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 0.

Step 1: Matrix Identification

To find the determinant of a matrix, first identify the elements of the 2x2 matrix: \[\begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} 0.2 & 0.4 \ -0.4 & -0.8 \end{bmatrix}\] where \( a = 0.2 \), \( b = 0.4 \), \( c = -0.4 \), and \( d = -0.8 \).

Step 2: Determinant Formula Application

For a 2x2 matrix, the determinant is calculated using the formula \(ad - bc\). Substitute the identified values into this formula:\[\det(A) = (0.2)(-0.8) - (0.4)(-0.4)\] where \(a = 0.2\), \(b = 0.4\), \(c = -0.4\), \(d = -0.8\).

Step 3: Calculate Each Product

Calculate each product individually:1. The product of \( a \) and \( d \):\(0.2 \times -0.8 = -0.16\).2. The product of \( b \) and \( c \):\(0.4 \times -0.4 = -0.16\).

Step 4: Determinant Calculation

Use the results of the products to find the determinant:\[\det(A) = -0.16 - (-0.16) = -0.16 + 0.16 = 0\].

Key Concepts

2x2 Matrix

A 2x2 matrix is a small-sized matrix that consists of two rows and two columns. Understanding this basic form of a matrix is essential as it helps in exploring more complex matrix operations. In a 2x2 matrix, the structure is represented as follows: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] where each of the four positions within the matrix is filled with specific numbers or variables called matrix elements.
These numbers could represent a variety of data points, such as coefficients in equations, values in statistics, or even simple numerical entries in computational tasks.
As a straightforward structure, the 2x2 matrix is an ideal starting point for learning about more intricate matrices and associated operations like finding determinants.

Matrix Elements

Matrix elements are the individual numbers within the rows and columns of a matrix. Each element in a matrix is identified by its position, usually denoted in terms of rows and columns. In a typical 2x2 matrix, the elements are labeled as \(a\), \(b\), \(c\), and \(d\).
Let's break it down:

• \(a\) refers to the element in the first row, first column.
• \(b\) is located in the first row, second column.
• \(c\) can be found in the second row, first column.
• \(d\) sits in the second row, second column.

Matrix elements are crucial because they act as the building blocks for performing calculations on matrices, such as finding totals, scaling, and computing determinants.
By manipulating these elements, one can transform matrices for various applications, ranging from computer graphics to solving systems of linear equations.

Determinant Calculation

Calculating the determinant of a 2x2 matrix is an essential task that helps determine properties like invertibility and solutions to systems of equations. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant, denoted as \( \det(A) \), is calculated using the formula: \( ad - bc \).
Here's how it's done using the example matrix:

• First, multiply the element \(a\) by the element \(d\): \( a \times d \).
• Next, multiply \(b\) by \(c\): \( b \times c \).
• Then, subtract the product of \(b\) and \(c\) from the product of \(a\) and \(d\): \( ad - bc \).

In this case, with \(a = 0.2\), \(b = 0.4\), \(c = -0.4\), and \(d = -0.8\):

• Compute \(0.2 \times -0.8 = -0.16\).
• Compute \(0.4 \times -0.4 = -0.16\).
• Finally, subtract the two: \(-0.16 - (-0.16) = 0\).

This example illustrates how a determinant of zero indicates that the matrix is singular, meaning it cannot be inverted, a vital concept in matrix algebra and its applications.