Problem 18 - Advanced Engineering Mathematics

Question

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{rr} 2 & -3 \\ -2 & 4 \end{array}\right) $$

Step-by-Step Solution

Answer

The inverse is \( \begin{pmatrix} 2 & 1.5 \\ 1 & 1 \end{pmatrix} \).

Step 1: Check Matrix Size

The given matrix is a 2x2 matrix. The inverse of a matrix can only exist if it is a square matrix, meaning it has the same number of rows and columns. Since this matrix is square (2x2), we can proceed to find its inverse if it exists.

Step 2: Calculate the Determinant

For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as \( ad - bc \). For the given matrix \( \begin{pmatrix} 2 & -3 \ -2 & 4 \end{pmatrix} \), the determinant is \( 2 \times 4 - (-3) \times (-2) = 8 - 6 = 2 \). Since the determinant is not zero, the inverse exists.

Step 3: Use the Determinant to Find the Inverse

The inverse of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated as \( \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \). Substitute \( a = 2 \), \( b = -3 \), \( c = -2 \), and \( d = 4 \) into this formula to find the inverse.

Step 4: Write the Inverse Matrix

The inverse matrix is calculated as \( \frac{1}{2} \begin{pmatrix} 4 & 3 \ 2 & 2 \end{pmatrix} \), which simplifies to \( \begin{pmatrix} 2 & 1.5 \ 1 & 1 \end{pmatrix} \). This means the inverse of the given matrix is: \[\begin{pmatrix}2 & 1.5 \1 & 1 \end{pmatrix}\]

Key Concepts

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix.
For a 2x2 matrix, the formula is simple and helpful in determining if an inverse exists.
To calculate the determinant of a 2x2 matrix: - Consider the matrix in the form \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \). - The determinant is computed as \( ad - bc \). This determinant tells us important information: - If the determinant is zero, the matrix does not have an inverse. - If it's non-zero, then the matrix has an inverse.
In our example, the determinant was computed as \( 2 \times 4 - (-3) \times (-2) = 8 - 6 = 2 \).
Since 2 is not zero, the matrix has an inverse.

2x2 Matrix

A 2x2 matrix is a simple structure that you may encounter in linear algebra. It's important in various mathematical and real-world applications.
- This type of matrix has 2 rows and 2 columns, resulting in a square structure.
The arrangement of elements in a 2x2 matrix can be: \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \].
The simplicity of a 2x2 matrix makes it a perfect starting point for learning more complex matrix operations. Here's why it's essential: - Easier computations: With fewer entries than larger matrices, operations like finding determinants or inverses are quicker. - Foundational concept: Understanding operations with 2x2 matrices helps you tackle larger matrices later.
Recognizing the structure of a 2x2 matrix will guide you in correctly applying algebraic operations.

Matrix Algebra

Matrix algebra is a powerful tool in mathematics and beyond. It encompasses all operations involving matrices and can be used to solve complex problems.
A key operation within matrix algebra is finding an inverse. The inverse of a matrix is like its opposite: - When a matrix is multiplied by its inverse, it results in the identity matrix.
For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the inverse is determined using: - The formula: \( \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \), where \( ad - bc \) is non-zero. Understanding how this formula works helps in matrix problem-solving.
Matrix algebra is not limited to finding inverses; it also includes operations like: - Multiplication, addition, and subtraction of matrices. - Solving systems of linear equations.
Mastering these concepts opens up new doors in mathematics, engineering, physics, and computer science.