Problem 12 - Precalculus: Functions and Graphs

Question

Find the determinant of the matrix. $$\left[\begin{array}{rr} c & d \\ -d & c \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is \( c^2 + d^2 \).

Step 1: Identify the Components of the 2x2 Matrix

We have the matrix \( \begin{bmatrix} c & d \ -d & c \end{bmatrix} \). This is a 2x2 matrix, where the elements are positioned as follows: \( a_{11} = c \), \( a_{12} = d \), \( a_{21} = -d \), and \( a_{22} = c \).

Step 2: Apply the Determinant Formula for a 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). For the given matrix, substitute in the corresponding values: \( (c)(c) - (d)(-d) \).

Step 3: Calculate Each Component of the Determinant

Calculate \( c \times c = c^2 \) and \( d \times (-d) = -d^2 \).

Step 4: Simplify the Determinant Expression

Substitute the calculated values into the determinant formula: \( c^2 - (-d^2) = c^2 + d^2 \).

Step 5: Conclude the Determinant Calculation

The determinant of the matrix \( \begin{bmatrix} c & d \ -d & c \end{bmatrix} \) is \( c^2 + d^2 \).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple yet fundamental unit in matrix algebra. It consists of two rows and two columns, creating a grid of four elements. These elements are usually denoted as \( a_{11}, a_{12}, a_{21}, \text{ and } a_{22} \). The position of each number or variable within the matrix is crucial for matrix operations. For example, in our original exercise, the given matrix is \( \begin{bmatrix} c & d \ -d & c \end{bmatrix} \). Here, each element is positioned uniquely:

• \( a_{11} = c \)
• \( a_{12} = d \)
• \( a_{21} = -d \)
• \( a_{22} = c \)

Understanding the structure is essential as it lays the groundwork for finding the determinant and performing other matrix operations. A 2x2 matrix is often used in various mathematical calculations, including transformations and systems of equations.

Determinant Formula

The determinant of a matrix serves as a key concept in matrix algebra, providing insights into the matrix's properties, such as whether it is invertible. For a 2x2 matrix, the determinant allows us to calculate a single number that reflects certain characteristics of the matrix.
In the context of a 2x2 matrix like \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \). This formula multiplies the elements of the diagonal from the top left to bottom right and subtracts the product of the elements from the other diagonal.
Applying this to our specific matrix \( \begin{bmatrix} c & d \ -d & c \end{bmatrix} \), we find the determinant by computing:

• \( c \times c = c^2 \)
• \( d \times (-d) = -d^2 \)

Substituting these into the determinant formula gives us \( c^2 + d^2 \). This result provides a scalar value representing the matrix's geometry.

Matrix Algebra

Matrix algebra involves various operations applied to matrices, including addition, subtraction, scalar multiplication, and finding determinants. In the scope of a 2x2 matrix, these operations become the building blocks for solving more complex problems in higher dimensions.
The determinant is particularly useful in matrix algebra because it helps determine if a matrix is invertible (a crucial property for solving linear equations). A 2x2 matrix is invertible if its determinant is non-zero.
Furthermore, matrix algebra is widely used in fields like physics, engineering, and computer science, where solving systems of equations and performing linear transformations is frequently necessary. Understanding how to operate with a 2x2 matrix efficiently is important for leveraging these applications. The determinant operation, as demonstrated, is a valuable technique for achieving this.

Precalculus

In precalculus, students often encounter matrices as a way to explore systems of equations and linear transformations. These concepts provide a foundational understanding that bridges the gap to more advanced topics in calculus and linear algebra.
Understanding the determinant of a 2x2 matrix is a key skill in precalculus. It gives students insight into how different matrix properties can affect solutions to equations and transformations.

• An invertible matrix indicates that a unique solution exists for the associated system of equations.
• Determinants also help in understanding translations, rotations, and scaling transformations in a plane.

By mastering these concepts, students prepare for the complexities of calculus, where they encounter matrices in contexts such as Jacobians, multiple integrals, and vector spaces. Hence, learning to compute determinants like \( c^2 + d^2 \) provides important mathematical maturity.