Problem 11 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix. $$ \left(\begin{array}{rr} 3 & 5 \\ -1 & 4 \end{array}\right) $$

Step-by-Step Solution

Answer

17

Step 1: Understand the Matrix

We are given a 2x2 matrix: \(\begin{pmatrix} 3 & 5 \ -1 & 4 \end{pmatrix}\). Our task is to calculate the determinant of this matrix.

Step 2: Recall the Determinant Formula for 2x2 Matrices

The formula for finding the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is \(ad - bc\).

Step 3: Apply the Determinant Formula

Substitute the values from the matrix into the determinant formula where \(a = 3\), \(b = 5\), \(c = -1\), and \(d = 4\). Calculate \(ad - bc\).\[\text{Determinant} = (3)(4) - (5)(-1)\]

Step 4: Calculate the Products

Calculate the products from Step 3. \[(3)(4) = 12\]\[(5)(-1) = -5\]

Step 5: Subtract the Products

Now subtract the product of \(bc\) from the product of \(ad\).\[12 - (-5) = 12 + 5 = 17\]

Step 6: State the Determinant

The determinant of the matrix is 17.

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is a simple way to organize numbers in mathematics, specifically in a table format with two rows and two columns. Each element in the matrix has its own position defined by its row and column number.For example, in the matrix \( \begin{pmatrix} 3 & 5 \ -1 & 4 \end{pmatrix} \), the number 3 is at the first row, first column, while 5 is at the first row, second column. The numbers -1 and 4 occupy the second row's first and second columns respectively.These matrices are fundamental building blocks in mathematics, particularly in linear algebra. They can represent various systems of linear equations, transformations in a plane, or even data points for analysis. Understanding their structure makes it easier to manipulate them and solve related problems effectively.

Determinant Formula for a 2x2 Matrix

The determinant of a 2x2 matrix provides important information about the properties of the matrix. The determinant for a matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using a specific formula:\[ \text{Determinant} = ad - bc \]This simple formula multiplies the elements of the main diagonal (from top-left to bottom-right) and subtracts the product of the elements of the off-diagonal (from top-right to bottom-left). For example, to find the determinant of the matrix \( \begin{pmatrix} 3 & 5 \ -1 & 4 \end{pmatrix} \), apply the formula: - Multiply elements along the main diagonal: \(3 \times 4 = 12\) - Multiply elements along the off-diagonal: \(5 \times (-1) = -5\)Subtract these results to get the determinant: \[ 12 - (-5) = 12 + 5 = 17 \]This calculation helps in determining properties such as the invertibility of the matrix or the volume scaling factor in transformations.

Matrix Calculation Method

Calculating the determinant involves basic arithmetic operations applied to the matrix's elements. Let's break down the steps in more detail for clarity:1. **Identify Matrix Elements** - Recognize and label each element of the matrix as \( a, b, c, \) and \( d \).2. **Apply Determinant Formula** - Use the formula \( ad - bc \) to combine these elements' values correctly.3. **Perform Multiplication** - Calculate the two products: \( a \times d \) and \( b \times c \). - Ensure accuracy in positive or negative signs especially when handling negative numbers.4. **Calculate Final Determinant Value** - Subtract the product of the off-diagonal from the main diagonal's product.Understanding and methodically following these steps allows you to accurately compute the determinant for any 2x2 matrix, reinforcing concepts like matrix operations and relevance in equations systems.

Introduction to Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, vector spaces (or linear spaces), and linear equations, among others. It is incredibly important because it provides techniques and methods to solve systems of linear equations, analyze geometric transformations, and understand space dimensions. Matrices, such as the 2x2 matrix discussed, are a key tool in linear algebra. They help perform and simplify operations, such as rotations and translations, and solve linear equations logically and efficiently. The determinant, specifically, gives insight into the properties of matrices, such as whether systems of equations have unique solutions or how transformations scale volumes in spaces. In studying linear algebra, understanding matrices and their determinants can lead to deeper comprehension of complex systems, simplifying many real-world problems in engineering, physics, computer science, and more.