Matrix Inverse - Learnexams

Matrix Inverse
Write a Python program to find the inverses of user-generated 2×2 matrices.
1) Do NOT use any packages (e.g., NumPy, pandas).
2) Ask the user to enter a matrix in the nested list expression.
3) Construct and display its inverse matrix. Ignore cases where the inverse does not exist.
The program should be capable of the following.
Enter 4 numbers separated by comma (a,b,c,d): 5,3,6,4 The inverse of the matrix is [[2.0, -1.5], [-3.0, 2.5)).
Follow the same submission requirement (.py file and screenshot) as before.
Note: the 4 input numbers a,b,c,d correspondes to the 2×2 matrix (ab).
Background: A matrix is an array of numbers, like a spreadsheet filled with numbers. It has wide applications in science, engineering, and businesses. The simplest form of matrices is a 2 by 2 matrix, e.g.. (3). In Python, we can use a nested list to express the matrix as shown above.
The inverse of a matrix A, typically denoted by A¹, is another matrix that satisfies AA¹ = AA(10). This is analogous to 2-2121-21 for real numbers. Matrix inverses can be used to solve systems of linear equations, among other things.
Calculation:
Let A = (ab). Its inverse is
A-ad-be(4)
Notice that when ad be, the inverse does not exist. But you can ignore this case for the assignment.
As an example, if A = (3), we have
A-15×4-6×3(653)=(3)
There are other (more efficient) ways to calculate matrix inverses.

The Correct Answer and Explanation is :

Python Program to Find the Inverse of a 2×2 Matrix

Here is the Python program that calculates the inverse of a 2×2 matrix. The user inputs the values in a nested list format (a, b, c, d), and the program calculates and prints the inverse matrix. If the inverse doesn’t exist (i.e., when the determinant is 0), the program ignores that case as per the assignment.

Python Code:

# Function to calculate the determinant of the matrix
def determinant(matrix):
    return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0]

# Function to calculate the inverse of a 2x2 matrix
def inverse(matrix):
    det = determinant(matrix)
    if det != 0:
        # Formula to find the inverse of a 2x2 matrix
        inverse_matrix = [
            [matrix[1][1] / det, -matrix[0][1] / det],
            [-matrix[1][0] / det, matrix[0][0] / det]
        ]
        return inverse_matrix
    else:
        return None  # Return None if inverse doesn't exist

# Input: User enters 4 values
input_values = input("Enter 4 numbers separated by comma (a,b,c,d): ").split(',')
a, b, c, d = float(input_values[0]), float(input_values[1]), float(input_values[2]), float(input_values[3])

# Create matrix from input
matrix = [[a, b], [c, d]]

# Calculate and display inverse
inv_matrix = inverse(matrix)
if inv_matrix:
    print(f"The inverse of the matrix is {inv_matrix}")
else:
    print("The inverse of the matrix does not exist.")

Explanation of the Code:

Determinant Calculation: The determinant of a 2×2 matrix A = [[a, b], [c, d]] is calculated as ad - bc. The determinant determines whether the matrix has an inverse or not. If the determinant is zero, the matrix does not have an inverse.
Inverse Formula: The formula for the inverse of a 2×2 matrix is given by:
[
A^{-1} = \frac{1}{{ad – bc}} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}
]
This formula works only when the determinant is non-zero. If the determinant is zero, the matrix does not have an inverse.
Program Flow:

The program asks the user to input four numbers corresponding to the elements of the matrix in the form a, b, c, d.
The matrix is created using these values and passed to the inverse function.
If the determinant is non-zero, the program calculates the inverse matrix using the formula and prints it.
If the determinant is zero, it informs the user that the inverse does not exist.

Example:

For input 5, 3, 6, 4, the matrix would be:

[
\begin{bmatrix}
5 & 3 \
6 & 4
\end{bmatrix}
]

The determinant is 5*4 - 3*6 = 20 - 18 = 2. Since the determinant is non-zero, the inverse is:

[
A^{-1} = \frac{1}{2} \begin{bmatrix} 4 & -3 \ -6 & 5 \end{bmatrix} = \begin{bmatrix} 2.0 & -1.5 \ -3.0 & 2.5 \end{bmatrix}
]

Output:

Enter 4 numbers separated by comma (a,b,c,d): 5,3,6,4
The inverse of the matrix is [[2.0, -1.5], [-3.0, 2.5]]

Image Generation:

Now, I will create a diagram of how a 2×2 matrix looks and its inverse for clarity. Let me generate that for you.

Here is the diagram showing a 2×2 matrix and its inverse. It includes the matrix with labeled elements (a, b, c, d) and the calculation of its inverse. This should help you understand how the formula works for finding the inverse. Let me know if you need any further clarification!