Problem 25 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The identity is verified: both sides equal \(ad - bc\).

Step 1: Understand the Determinants

First, recall the formula to calculate a 2x2 determinant. For a matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is calculated as \(ad - bc\). We'll use this in the next steps to evaluate each determinant.

Step 2: Expand the Determinant on the Left Side

Use the determinant formula on the left side. For \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), calculate the determinant as \(ad - bc\).

Step 3: Expand the Determinant on the Right Side

Now apply the determinant formula to the right-side matrix \(\begin{pmatrix} a & b \ ka+c & kb+d \end{pmatrix}\). The determinant is \(a(kb+d) - b(ka+c)\), which simplifies to \(akb + ad - bka - bc\).

Step 4: Simplify the Right Side Expression

Observe that in the expression \(akb + ad - bka - bc\), the terms \(akb\) and \(-bka\) cancel each other out as they are equal and opposite. The remaining terms are \(ad - bc\).

Step 5: Verify the Identity

Compare the simplified determinant of the right side, \(ad - bc\), to the determinant of the left side, \(ad - bc\). Since both are equal, the identity is verified.

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple, square arrangement of numbers in a grid with two rows and two columns. This type of matrix is often written in the form \[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]where \(a\), \(b\), \(c\), and \(d\) are elements or values placed in each position of the grid. Each element has a specific place: \(a\) and \(b\) are in the first row, and \(c\) and \(d\) are in the second row.
Understanding the layout and structure of a 2x2 matrix is crucial in mathematics, especially in linear algebra, as it sets the foundation for more complex concepts and computations, such as finding determinants and using matrices in systems of equations.

Determinant Formula

The determinant of a 2x2 matrix gives valuable information about the matrix. It is often used to determine properties like whether a matrix can be inverted or how transformations affect areas and volumes. For a matrix set up as \[\begin{pmatrix} a & b \ c & d \end{pmatrix},\]the determinant is calculated using the formula:\[\text{Determinant} = ad - bc.\]
This simple equation multiplies the diagonal elements (\(a\) and \(d\)) and subtracts the product of the off-diagonal elements (\(b\) and \(c\)).

• \(ad\) and \(bc\) terms are crucial because they directly influence the property of the matrix.
• If the determinant is zero, the matrix is considered singular and non-invertible.
• A non-zero determinant indicates an invertible matrix.

The determinant allows us to understand key characteristics of the matrix relating to its linear transformations.

Identity Verification

Identity verification through determinants involves checking if two matrices express the same transformation. In this context, the task is to prove whether both sides of the given identity share equal determinants.
To verify the identity:

• Firstly, expand the determinant of each matrix using the determinant formula.
• On the left side, calculate the determinant of the matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) to get \(ad - bc\).
• On the right side, the matrix is \(\begin{pmatrix} a & b \ ka+c & kb+d \end{pmatrix}\), and expanding it gives \(a(kb+d) - b(ka+c)\).
• Simplifying this expression by cancellation of terms results in \(ad - bc\), confirming both sides are equal.

The equality \(ad - bc\) on both sides confirms the identity, thus verifying that both matrices have equivalent determinants and represent the same transformation.