Problem 72 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

After evaluating both sides of the equation, it is found that LHS = RHS, thus the original equation holds true.

Step 1: Evaluate the Left-hand Side (LHS)

The left side of the equation is the determinant of a 2x2 matrix. It is evaluated as follows: \( \left| \begin{array}{c c} w & cx \ y & cz \end{array} \right| = (wcz) - (ycx) = c(wz - yx)\)

Step 2: Evaluate the Right-hand Side (RHS)

The right side of the equation is the scalar multiple of another determinant. It is evaluated as follows: \( c \left| \begin{array}{c c} w & x \ y & z \end{array} \right| = c((wz) - (yx)) = c(wz - yx)\)

Step 3: Compare LHS and RHS

Now that we have simplified both the LHS and the RHS, we can see that they are equal. Hence, the original equation holds true.

Key Concepts

Determinant of a Matrix

Understanding the concept of a determinant is crucial because it provides insight into the properties of a matrix. For a 2x2 matrix, the determinant can be calculated using a simple formula: Given a matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), its determinant, denoted as \( |A| \), is computed as \( ad - bc \). In the context of the exercise, the determinant of the matrix with elements \( w, cx, y, cz \) is obtained by multiplying \(w \) and \( cz \) and then subtracting the product of \( y \) and \( cx \).

This calculation is a fundamental procedure in linear algebra, as it helps in understanding whether a system of linear equations has a unique solution or not. The determinant also plays a key role in other operations like calculating the inverse of a matrix and in understanding geometric properties of linear transformations.

Properties of Determinants

Determinants exhibit fascinating and useful properties which make solving mathematical problems easier. A key property, which is applied in the exercise provided, is that the determinant of a matrix is linear in each row and column. This means that if you multiply a row or column by a scalar, the determinant of the matrix is multiplied by the same scalar. Another important property is the determinant of a matrix remains the same if a row (or column) is added to another row (or column) multiplied by a scalar.

There are more characteristics that can be highly useful: the determinant changes sign if two rows (or two columns) are swapped, and the determinant of an identity matrix is always one. Understanding these properties is not just academic; they enable us to solve complex equations and understand multidimensional space in a much simpler way.

Scalar Multiplication of Determinants

Scalar multiplication of determinants is a straight-forward process, yet it's a powerful tool in matrix algebra. When you multiply a scalar value by a determinant, you effectively scale every element of a single row or column by that scalar. As per the properties of determinants, this operation scales up the whole determinant value by the same factor.

For example, if you have a 2x2 matrix \( A \) and you want to multiply it by a scalar \( c \), every element of \( A \) gets multiplied by \( c \) which, when you take the determinant, translates to the determinant \( |A| \) being multiplied by \( c \) as well. This procedure simplifies many mathematical scenarios, especially in the step by step solution provided in the exercise.

2x2 Matrix

When it comes to the simplicity of calculating determinants, the 2x2 matrix fits the bill perfectly. It is the smallest square matrix from which a non-zero determinant can be calculated, and its computation is easy to grasp. A 2x2 matrix \( A \) takes the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), with its determinant calculated as \( ad - bc \).

Exercise Application

The exercise provided showcases how determinants of a 2x2 matrix are evaluated and how straightforward the process is. Here, by placing the values \( w, x, y, z, \) and their scalar multiples into this simple formula, we can understand the relationships between the elements of the matrix, and verify the given equation.

As you study linear algebra, you'll find that the 2x2 matrix often serves as a building block for understanding more complex matrix operations and determinants.