Problem 10 - Precalculus: Functions and Graphs

Question

Without expanding, explain why the statement is true. $$\left|\begin{array}{rr} 2 & -2 \\ 1 & 1 \end{array}\right|=-\left|\begin{array}{rr} -2 & 2 \\ 1 & 1 \end{array}\right|$$

Step-by-Step Solution

Answer

Swapping columns changes the sign of the determinant, so \(|A| = -|B|\).

Step 1: Understand Determinants and Properties

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( ad - bc \). A key property of determinants is that swapping the columns of a matrix changes the sign of its determinant.

Step 2: Identify Matrices and Their Determinants

Let's identify the two matrices: \( A = \begin{bmatrix} 2 & -2 \ 1 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} -2 & 2 \ 1 & 1 \end{bmatrix} \). We need to determine \( |A| \) and \( |B| \) and then show why \( |A| = -|B| \) without explicitly calculating the determinants yet.

Step 3: Use Swapping Property of Determinants

Notice that matrix \( B \) can be obtained from matrix \( A \) by swapping the positions of the first column with columns adjusted by multiplying by -1. This operation changes the sign of the determinant according to determinant properties. Thus, \( |A| = -|B| \).

Key Concepts

2x2 matrices

A 2x2 matrix is a simple yet powerful structure in linear algebra. It consists of two rows and two columns, forming a grid with four elements. For example, consider the 2x2 matrix:\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]Each element in this matrix has a specific position. The matrix can be thought of as a little box where each element impacts the calculations in a fundamental way.They are small, but they pack a lot of information, acting as a foundation for understanding larger matrix operations. Because they have only four elements, calculations, such as finding the determinant, are straightforward, which makes them an excellent starting point for learning about matrices.

properties of determinants

The determinant is a special number associated with square matrices, which, for a 2x2 matrix, is calculated as follows:\[ |A| = ad - bc\]This formula reflects its simplicity yet importance in matrix algebra. Few notable properties of determinants are:

• Changes in the order of matrix rows or columns can affect the determinant's sign.
• If two rows or columns of a matrix are identical, the determinant is zero.
• Swapping two rows or two columns flips the sign of the determinant.

Understanding these properties allows us to explore deeper concepts, such as linear dependence and the ability of the matrix to be inverted. These characteristics make determinants essential in solving systems of linear equations and understanding matrix transformations.

matrix column swapping

Swapping columns in a matrix is a straightforward operation that has interesting effects on the determinant. Specifically, when you swap two columns in a matrix, it changes the sign of the determinant due to one of its core properties. Consider matrix \( A \) and the matrix \( B \), obtained by swapping columns in \( A \):\[ A = \begin{bmatrix} 2 & -2 \ 1 & 1 \end{bmatrix} \]\[ B = \begin{bmatrix} -2 & 2 \ 1 & 1 \end{bmatrix} \]Matrix \( B \) can be seen as matrix \( A \) with its columns flipped and adjusted by multiplying by -1. This swap leads to the determinant of \( B \) being the negative of the determinant of \( A \), thus, \(|A| = -|B|\). This arithmetic flip is an essential aspect of the determinant's behavior, reflecting its sensitivity to changes in the matrix's structure.