Problem 12 - Advanced Engineering Mathematics

Question

In Problems 9-14, evaluate the determinant of the given matrix. $$ \left(\begin{array}{rr} \frac{1}{4} & \frac{1}{2} \\ \frac{1}{3} & -\frac{4}{3} \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is \(-\frac{1}{2}\).

Step 1: Understand the determinant formula for 2x2 matrix

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula: \( ad - bc \). This formula will be applied to our specific matrix.

Step 2: Identify the elements of the matrix

In the given matrix \( \begin{pmatrix} \frac{1}{4} & \frac{1}{2} \ \frac{1}{3} & -\frac{4}{3} \end{pmatrix} \), identify the elements as: \( a = \frac{1}{4} \), \( b = \frac{1}{2} \), \( c = \frac{1}{3} \), and \( d = -\frac{4}{3} \).

Step 3: Multiply diagonals according to the formula

Calculate the products of the diagonals: \( ad = \left(\frac{1}{4}\right)\left(-\frac{4}{3}\right) = -\frac{1}{3} \)and \( bc = \left(\frac{1}{2}\right)\left(\frac{1}{3}\right) = \frac{1}{6} \).

Step 4: Subtract the products

Use the determinant formula to subtract the products: \( determinant = ad - bc = -\frac{1}{3} - \frac{1}{6} \).

Step 5: Simplify the expression

Combine the fractions: \( ad - bc = -\frac{1}{3} - \frac{1}{6} = -\frac{2}{6} - \frac{1}{6} = -\frac{3}{6} = -\frac{1}{2} \). Therefore, the determinant of the matrix is \(-\frac{1}{2}\).

Key Concepts

2x2 matrix

Let's begin by diving into what a 2x2 matrix really is. A 2x2 matrix is one of the simplest kinds of matrices you'll encounter in linear algebra. It consists of two rows and two columns, creating a small square-like structure of numbers.
Each element in the matrix can be a number, fraction, or a symbol such as a variable. For example, the matrix given in this exercise is:

• a = \(\frac{1}{4}\)
• b = \(\frac{1}{2}\)
• c = \(\frac{1}{3}\)
• d = \(-\frac{4}{3}\)

These elements are always written in a specific order: the first row is \([a, b]\) and the second row is \([c, d]\). Remember, the way we position numbers matters greatly in a matrix.
In mathematical terms, this setup can be expressed as: \[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]

linear algebra

Linear algebra is a branch of mathematics dealing with vectors and matrices. It's immensely helpful in solving systems of linear equations and understanding transformations in coordinate space.
Matrices, like our 2x2 matrix, are fundamental components of linear algebra as they can effortlessly handle multiple equations and model complex systems. They serve as a way to organize and manipulate data.
When we calculate determinants, we're uncovering properties of these matrices, such as whether they can be inverted or what kind of geometric transformation they might represent. The determinant gives us crucial insights into the matrix's behavior, especially in applications like computer graphics, engineering, and data science.
Overall, linear algebra allows you to work with data on both small and large scales, making it a vital tool in both theoretical and applied settings.

determinant calculation

Calculating the determinant is one of the most important techniques used in linear algebra. Determinants have vital applications, such as finding the inverse of a matrix, solving linear systems, and determining matrix properties.
For a 2x2 matrix, the determinant gives us a single number derived from the specific arrangement of the matrix elements. The formula to calculate the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is straightforward:

• Multiply a by d (these elements lie on the main diagonal).
• Subtract the product of b and c (elements on the opposite diagonal).

Thus, the formula looks like this: \[ ad - bc \] In our exercise, we plug in the values: \((\frac{1}{4})(-\frac{4}{3}) - (\frac{1}{2})(\frac{1}{3}) \)These operations resulted in the determinant \(-\frac{1}{2}\), a negative value which typically indicates the transformation flips orientation in space.