Problem 15 - Advanced Engineering Mathematics

Question

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{rr} 6 & -2 \\ 0 & 4 \end{array}\right) $$

Step-by-Step Solution

Answer

The inverse is \( \begin{pmatrix} \frac{1}{6} & \frac{1}{12} \\ 0 & \frac{1}{4} \end{pmatrix} \).

Step 1: Calculate the Determinant

For a matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated as \( ad - bc \). Here, for the matrix \( \begin{pmatrix} 6 & -2 \ 0 & 4 \end{pmatrix} \), the determinant is \( 6 \times 4 - (-2) \times 0 = 24 \). Since the determinant is not zero, the inverse exists.

Step 2: Apply the Formula for Inverse of a 2x2 Matrix

The formula for the inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is given by \( A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \). With \( a = 6 \), \( b = -2 \), \( c = 0 \), and \( d = 4 \), the inverse of the matrix is \( \frac{1}{24} \begin{pmatrix} 4 & 2 \ 0 & 6 \end{pmatrix} \).

Step 3: Simplify the Inverse Matrix

Multiply each element of the matrix \( \begin{pmatrix} 4 & 2 \ 0 & 6 \end{pmatrix} \) by \( \frac{1}{24} \). This results in the matrix \( \begin{pmatrix} \frac{1}{6} & \frac{1}{12} \ 0 & \frac{1}{4} \end{pmatrix} \). Therefore, the inverse of the given matrix is: \[ \begin{pmatrix} \frac{1}{6} & \frac{1}{12} \ 0 & \frac{1}{4} \end{pmatrix} \].

Key Concepts

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. It gives important information about matrices, such as whether functions have an inverse. When it comes to a 2x2 matrix, the determinant also helps to determine if the matrix is invertible. For a matrix \[A = \begin{pmatrix} a & b \ c & d \end{pmatrix},\]the determinant can be calculated using the formula:

• Determinant = \( ad - bc \)

To find the determinant of the matrix \[\begin{pmatrix} 6 & -2 \ 0 & 4 \end{pmatrix},\]substitute the values: \( a = 6 \), \( b = -2 \), \( c = 0 \), and \( d = 4 \). The calculation will be:\[6 \times 4 - (-2) \times 0 = 24\]Since the result is 24, which is not zero, it means the matrix has an inverse. Calculating the determinant is crucial because a zero determinant would mean the matrix does not have an inverse.

Matrix Algebra

Matrix algebra is a fascinating and essential area of mathematics focusing on matrices and their operations. Some of the key operations include addition, subtraction, multiplication, and finding inverses of matrices. These operations follow specific rules and are essential in various applications across mathematics and science.
Mathematics often requires manipulating matrices to solve systems of equations, transform geometric figures, and model real-world scenarios. A matrix represents data in a format that can be handled efficiently, and understanding matrix algebra is fundamental to dealing efficiently with such datasets.
Although matrices can have any dimension, a 2x2 matrix is quite common due to its simplicity and practical applications. In matrix algebra, ensuring a matrix is invertible is vital, particularly for solving linear equations. To determine this, one must compute the determinant as an initial step. Matrix algebra helps us to employ inverses in calculations to find solutions that would otherwise be difficult to derive.

2x2 Matrix Inverse

Finding the inverse of a 2x2 matrix is a straightforward process once you have computed the determinant, assuming it is not zero. Having a non-zero determinant guarantees that the matrix can be inverted, which opens up many possibilities in solving linear problems.
For a 2x2 matrix \[A = \begin{pmatrix} a & b \ c & d \end{pmatrix},\]the formula for the inverse is:\[A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}.\]Applying this to our example matrix \[\begin{pmatrix} 6 & -2 \ 0 & 4 \end{pmatrix}\]gives us:

• First, confirm the determinant is 24.
• Substitute into the inversion formula: \( a = 6 \), \( b = -2 \), \( c = 0 \), \( d = 4 \).
• Plug these values into the inverse formula: \[ A^{-1} = \frac{1}{24} \begin{pmatrix} 4 & 2 \ 0 & 6 \end{pmatrix}. \]
• Simplify the resulting matrix by multiplying each element by \( \frac{1}{24} \).

This results in the final inverse matrix:\[\begin{pmatrix} \frac{1}{6} & \frac{1}{12} \ 0 & \frac{1}{4} \end{pmatrix}.\]The inverse matrix is a powerful tool, allowing transformations and solving equations that utilize matrix algebra.