Problem 74 - Precalculus with Limits

Question

In Exercises 71-76, evaluate the determinant(s) to verify the equation. \(\left| \begin{array}{r} w & x \\ cw & cx \end{array} \right| = 0\)

Step-by-Step Solution

Answer

After calculating and simplifying the expression, we find that the value of the determinant is indeed 0.

Step 1: Understand the Elements of the Determinant

The given determinant is \(\left| \begin{array}{r} w & x \ cw & cx \end{array} \right|\). Here, the elements of the determinant matrix are w, x in the first row and cw, cx in the second row. We will use these elements in the formula for evaluating the determinant.

Step 2: Apply the 2x2 determinant formula

The formula to find the determinant of a 2x2 matrix A = [\n[a, b],\n[c, d]] is det(A) = ad - bc. Apply this formula to the given determinant to get the result. So, the determinant is \(w \cdot cx - x \cdot cw\).

Step 3: Simplify the expression

Simplify \(w \cdot cx - x \cdot cw\) to get \(wcx - wcx\), which simplifies to 0.

Key Concepts

2x2 matrix

A 2x2 matrix is a foundational concept in linear algebra, crucial for calculating determinants. It represents a grid with two rows and two columns. Each value in the matrix is known as an element.For example, a 2x2 matrix can be denoted as: \[\begin{pmatrix} a & b \ c & d\end{pmatrix}\]In this visual representation:* The first row is \(a, b\)* The second row is \(c, d\)These elements can be any real number, variable, or expression. Working with a 2x2 matrix involves manipulating these elements through specific mathematical rules or operations like finding the determinant. This specific type of matrix is straightforward, making it ideal for those learning about matrices for the first time. Understanding the structure and interpretation of a 2x2 matrix is essential when engaging in more complex linear transformations or when solving linear systems.

determinant formula

The determinant of a matrix is a special number assigned to a square matrix. For a 2x2 matrix, this value is calculated using a simple formula, which helps in understanding properties like system solvability in linear equations.Consider a 2x2 matrix: \[\begin{pmatrix} a & b \ c & d\end{pmatrix}\]The determinant of this matrix is given by the formula:\[\text{det}(A) = ad - bc\]This formula involves multiplying the diagonal elements \(a\) and \(d\), then subtracting the product of the opposite diagonal elements \(b\) and \(c\). Despite its simplicity, calculating the determinant can reveal critical properties of the matrix. For instance, if the determinant is zero, it indicates that the matrix is singular, meaning it doesn’t have an inverse, or its associated linear equations are dependent. Understanding and applying this formula effectively is crucial for successfully assessing the characteristics of a matrix in various mathematical and applied contexts.

matrix simplification

Matrix simplification involves reducing an expression involving matrices to its simplest form. In the context of determinants, it means performing and simplifying algebraic operations to reach a final value.Taking the previously calculated determinant expression:\[w \cdot cx - x \cdot cw\]Here, simplification entails identifying like terms and reducing them. In this case, both terms result in the same product, \(wcx\), which means they cancel each other out:\[wcx - wcx = 0\]This step is critical as it provides the final, simplified result of the determinant evaluation. Simplification is often necessary to verify if equations or representations adhere to given mathematical properties or conditions, such as the equation being equal to zero, as demonstrated in this exercise. This clarity allows for easier interpretation and understanding of the results.