Problem 8 - Precalculus Mathematics for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} -2 & 1 \\ 3 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 1.

Step 1: Understand the Matrix

The matrix given is a 2x2 matrix:\[\begin{bmatrix}-2 & 1 \3 & -2\end{bmatrix}\]A 2x2 matrix is located in a determinant formula with a specific calculation method.

Step 2: Recall the Determinant Formula for 2x2 Matrix

For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula:\[det(A) = ad - bc\]This simple multiplication and subtraction can be directly applied to the matrix values.

Step 3: Plug Values into the Formula

Assign the matrix's elements to variables for calculation:\(a = -2\), \(b = 1\), \(c = 3\), and \(d = -2\).Now substitute these into the determinant formula:\[det(A) = (-2)(-2) - (1)(3)\]

Step 4: Calculate the Determinant

Perform the calculations:First, multiply the first diagonal: \((-2)(-2) = 4\).Next, multiply the second diagonal: \((1)(3) = 3\).Now, subtract these results: \[4 - 3 = 1\]So, the determinant of the matrix is 1.

Key Concepts

2x2 Matrix

Matrices are fundamental structures in mathematics, often used to solve linear equations, perform transformations, and represent data. A 2x2 matrix is a simple square matrix that consists of 2 rows and 2 columns. It takes the form: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]Where \(a, b, c,\) and \(d\) are elements of the matrix. In practical terms, think of these elements as slots that can hold numbers or variables. Understanding the layout of a 2x2 matrix is crucial for performing matrix operations.

• The elements in the first row are \(a\) and \(b\).
• The elements in the second row are \(c\) and \(d\).

The two-by-two dimension implies both rows and columns are of equal size making operations like finding determinants straightforward. In this exercise, you correctly identified the positions of each element in the matrix, which is a key step before performing calculations.

Determinant Formula

The determinant of a matrix is a special number that can provide insights into the matrix's properties, such as invertibility, or the volume scaling factor for transformations. For a 2x2 matrix, the determinant is calculated using a straightforward formula:\[ det(A) = ad - bc \]Here’s what each part represents:

• \(a\) and \(d\) are the elements on the main diagonal (top-left to bottom-right).
• \(b\) and \(c\) are the elements on the opposite diagonal (top-right to bottom-left).

The calculation involves a simple multiplication of these diagonals, and then subtracting the product of the secondary diagonal from the product of the main diagonal. This method relies on the particular arrangement and sign multiplication within the matrix structure.

Matrix Calculation Steps

Calculating the determinant of a 2x2 matrix involves a few clearly defined steps. These steps ensure that you derive and understand the result correctly. Let's break down the process:1. **Identify and Assign Values**: Extract the values from the matrix and assign them as variables. - In our matrix, these are: \(a = -2\), \(b = 1\), \(c = 3\), \(d = -2\).2. **Apply the Determinant Formula**: Use the formula for finding the determinant of a 2x2 matrix. - Plug the values into the formula: \[det(A) = (-2)(-2) - (1)(3)\]3. **Perform Calculations**: - Multiply the elements of the first diagonal: \((-2) \times (-2) = 4\) - Multiply the elements of the second diagonal: \(1 \times 3 = 3\) - Subtract the results of these products: \(4 - 3 = 1\)These steps decompose the process into manageable tasks, allowing you to follow through logical calculations and ensuring accuracy. By systematically following these steps, you effectively utilized the formula to find that the determinant of this matrix is indeed 1.