Problem 24 - Precalculus: Functions and Graphs

Question

Verify the Identity by expanding each determinant. $$\left|\begin{array}{ll} a & b \\ k c & k d \end{array}\right|=k\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|$$

Step-by-Step Solution

Answer

The identity is verified; both sides are equal.

Step 1: Write the Determinants

Write down the determinants given in the problem. First, the determinant on the left: \( \left| \begin{array}{cc} a & b \ kc & kd \end{array} \right| \). Second, the determinant on the right: \( \left| \begin{array}{cc} a & b \ c & d \end{array} \right| \).

Step 2: Expand the Left Determinant

The determinant on the left is expanded using the formula for a 2x2 matrix: \( ad - bc \). For the determinant \( \left| \begin{array}{cc} a & b \ kc & kd \end{array} \right| \), apply the formula: \( a(kd) - b(kc) = akd - bkc \).

Step 3: Expand the Right Determinant

Expand the determinant \( \left| \begin{array}{cc} a & b \ c & d \end{array} \right| \) using the same formula for a 2x2 determinant: \( ad - bc \).

Step 4: Multiply the Right Determinant by k

Multiply the expanded form of the right determinant by \( k \): \( k(ad - bc) = kad - kbc \).

Step 5: Compare Both Sides

Now, compare the two expressions we have: On the left, we have \( akd - bkc \), and on the right, we also have \( kad - kbc \). Both expressions are identical.

Key Concepts

Matrix Algebra

Matrix algebra is a fundamental concept in mathematics, especially in the fields of linear algebra and computer science. Working with matrices involves understanding various operations and properties that apply to matrix elements. One of these operations is finding the determinant, which is a special number assigned to a matrix.

In matrix algebra:

• A matrix is an array of numbers arranged in rows and columns.
• Matrices can be of different sizes, generally denoted as an "m x n" matrix, where "m" is the number of rows and "n" is the number of columns.
• Determinants provide useful information about matrices, such as whether the matrix has an inverse.
• While this might seem abstract, determinants and matrices are applied in various real-world problems like network analysis and solving systems of linear equations.

Understanding determinants is crucial because they play a key role in solving complex equations and transforming geometric shapes on a coordinate plane.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, consisting of two rows and two columns. Despite its simplicity, it lays the groundwork for understanding larger matrices in matrix algebra.

Here's the general structure of a 2x2 matrix:

• The top row contains two elements, denoted as "a" and "b".
• The bottom row contains two elements, denoted as "c" and "d".

A compact way to represent this matrix is:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]With a 2x2 matrix, you can perform basic operations like addition, subtraction, and the calculation of a determinant. The straightforward structure of a 2x2 matrix makes learning these operations a manageable first step in matrix algebra.

Determinant Expansion

Determinant expansion is the process used to calculate the determinant of a matrix, which is crucial in many mathematical applications. For a 2x2 matrix, the formula for its determinant is quite simple: \( ad - bc \).

To understand this, let's break down the formula for a 2x2 matrix:

• The determinant is computed by taking the product of the elements on the main diagonal (from top-left to bottom-right), which is \( ad \).
• Then, subtract the product of the elements on the secondary diagonal (from top-right to bottom-left), which is \( bc \).
• This results in the formula: \( ad - bc \).

Expanding a determinant, especially for larger matrices, involves more complex rules like the cofactor expansion. However, for our 2x2 matrix, the simplicity of \( ad - bc \) allows us to quickly compute the determinant.

Identity Verification

Identity verification in mathematics involves proving that two expressions are equal under the algebraic rules and operations applied. This is a significant exercise in verifying properties or theorems about matrices and determinants.

For example, when verifying the identity given in the exercise:

• The identity states that the determinant of a matrix with rows scaled by a factor \( k \) is equal to \( k \) times the determinant of the original matrix.
• By expanding both determinants and comparing them, as done in the exercise, we can verify their equality.
• This process not only reinforces the understanding of determinants but also illustrates linear algebra concepts in action.

Identity verification builds confidence in handling matrices and helps in understanding deeper truths about matrix operations and linear transformations.