Problem 12 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix. $$ \left(\begin{array}{lr} \frac{1}{4} & \frac{1}{2} \\ \frac{1}{3} & -\frac{4}{3} \end{array}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify Matrix Elements

The matrix is a 2x2 matrix, identified as \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), where \( a = \frac{1}{4} \), \( b = \frac{1}{2} \), \( c = \frac{1}{3} \), and \( d = -\frac{4}{3} \).

Step 2: Understand Determinant Formula for 2x2 Matrices

For a 2x2 matrix, the determinant is calculated using the formula: \( \text{det}(A) = ad - bc \).

Step 3: Substitute Values into the Determinant Formula

Substitute the identified matrix elements into the formula: \( \text{det}(A) = \left( \frac{1}{4} \right) \left( -\frac{4}{3} \right) - \left( \frac{1}{2} \right) \left( \frac{1}{3} \right) \).

Step 4: Calculate Product of Diagonal Elements

Calculate \( ad = \frac{1}{4} \times -\frac{4}{3} = -\frac{4}{12} = -\frac{1}{3} \).

Step 5: Calculate Product of Off-Diagonal Elements

Calculate \( bc = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \).

Step 6: Compute the Determinant

Use the values from previous steps: \( \text{det}(A) = -\frac{1}{3} - \frac{1}{6} = -\frac{2}{6} - \frac{1}{6} = -\frac{3}{6} = -\frac{1}{2} \).

Key Concepts

2x2 matrix

A 2x2 matrix is a simple form of a matrix used frequently in mathematics, especially in linear algebra. It consists of two rows and two columns. This small size makes it particularly useful for introductory examples. A generic 2x2 matrix is often written as \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \). Each element in the matrix is denoted by a letter, making up a set of numbers that can be used in calculations like determining a determinant. The first row contains elements \( a \) and \( b \), while the second row contains \( c \) and \( d \). By understanding each cell's position and value, it's easier to apply operations such as finding determinants. The matrix given in the exercise fits this description, showing four specific numbers arranged in these two rows and two columns.

matrix elements

Matrix elements are simply the individual numbers found within a matrix. For a 2x2 matrix, you have four elements. These elements usually have specific positions: the top left is \( a \), the top right is \( b \), the bottom left is \( c \), and the bottom right is \( d \). Knowing the exact positions helps when calculating various matrix properties.

In the given example, the elements are clear. They are:

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

These values are essential for further calculations, like figuring out the determinant. Each element must be precisely placed and valued to maintain the matrix structure, allowing accurate calculations.

determinant formula

The determinant is an important value computed from the elements of a square matrix. For a 2x2 matrix, the determinant follows a straightforward formula which is crucial for performing further operations like solving systems of equations. The formula is: \[\text{det}(A) = ad - bc\]This equation involves multiplying the elements of the main diagonal (\( a \) and \( d \)) and then subtracting the product of the other diagonal (\( b \) and \( c \)).

For the given matrix:

• The product of the main diagonal: \( \frac{1}{4} \times -\frac{4}{3} = -\frac{1}{3} \)
• The product of the other diagonal: \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \)

Plugging these products into the formula, we find: \[\text{det}(A) = -\frac{1}{3} - \frac{1}{6} = -\frac{1}{2}\]This result tells you about the linear independence and area scaling of the transformation described by the matrix.