What is the Solution of a System of Two Linear Equations using Matrices?

S7-SA2-0247

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Finding the solution of a system of two linear equations using matrices means figuring out the values for the unknown variables (like 'x' and 'y') that make both equations true, by using matrix operations. It's a structured way to solve multiple equations at once.

Simple Example

Quick Example

Imagine you buy 2 samosas and 1 chai for Rs 50. Your friend buys 1 samosa and 2 chai for Rs 40. We can write these as equations: 2x + y = 50 and x + 2y = 40, where 'x' is the price of a samosa and 'y' is the price of a chai. Using matrices helps us find 'x' and 'y' systematically.

Worked Example

Step-by-Step

Let's solve: 2x + y = 7 and x + 3y = 11

1. Write the equations in matrix form AX = B.
A = [[2, 1], [1, 3]] (coefficient matrix)
X = [[x], [y]] (variable matrix)
B = [[7], [11]] (constant matrix)

2. Find the determinant of matrix A (det(A)).
det(A) = (2 * 3) - (1 * 1) = 6 - 1 = 5.
--- If det(A) is 0, a unique solution doesn't exist. Here, det(A) = 5, so we can proceed.

3. Find the adjoint of matrix A (adj(A)).
adj(A) = [[3, -1], [-1, 2]] (swap diagonal elements, change sign of off-diagonal elements)

4. Calculate the inverse of matrix A (A^-1).
A^-1 = (1/det(A)) * adj(A) = (1/5) * [[3, -1], [-1, 2]] = [[3/5, -1/5], [-1/5, 2/5]]

5. Use the formula X = A^-1 * B to find X.
X = [[3/5, -1/5], [-1/5, 2/5]] * [[7], [11]]

6. Perform matrix multiplication.
x = (3/5 * 7) + (-1/5 * 11) = 21/5 - 11/5 = 10/5 = 2
y = (-1/5 * 7) + (2/5 * 11) = -7/5 + 22/5 = 15/5 = 3

Answer: So, x = 2 and y = 3.

Why It Matters

This method is super important for solving complex problems in AI/ML to train models, in Physics to calculate forces, and in Engineering to design structures. Engineers use it to make sure bridges are strong, and data scientists use it to understand large datasets, helping them build smarter apps and systems.

Common Mistakes

MISTAKE: Swapping elements incorrectly when finding the adjoint matrix. | CORRECTION: Remember to swap the main diagonal elements and change the signs of the off-diagonal elements.

MISTAKE: Forgetting to check if the determinant is zero before finding the inverse. | CORRECTION: Always calculate the determinant first. If det(A) = 0, the inverse does not exist, and there's no unique solution.

MISTAKE: Errors in basic arithmetic during matrix multiplication. | CORRECTION: Double-check each multiplication and addition step carefully, especially when dealing with fractions.

Practice Questions

Try It Yourself

QUESTION: Solve for x and y: x + y = 5, x - y = 1 | ANSWER: x = 3, y = 2

QUESTION: Solve for x and y using matrices: 3x + 2y = 12, x + y = 5 | ANSWER: x = 2, y = 3

QUESTION: Find the values of x and y for the system: 2x - 3y = 1, 4x + y = 9. Use the matrix method. | ANSWER: x = 2, y = 1

MCQ

Quick Quiz

For the system 2x + y = 5 and 4x + 2y = 10, what is the determinant of the coefficient matrix?

-1

The Correct Answer Is:

The coefficient matrix is [[2, 1], [4, 2]]. The determinant is (2 * 2) - (1 * 4) = 4 - 4 = 0. This means the system has infinitely many solutions or no solution.

Real World Connection

In the Real World

In FinTech, banks use matrix methods to manage large portfolios of loans and investments. For example, to calculate how different interest rates affect multiple loans simultaneously, complex systems of equations are formed and solved using matrices to assess risk and make financial predictions.

Key Vocabulary

Key Terms

MATRIX: A rectangular arrangement of numbers or expressions in rows and columns. | DETERMINANT: A special number calculated from a square matrix, indicating if a unique solution exists. | INVERSE MATRIX: A special matrix that, when multiplied by the original matrix, results in an identity matrix. Used to 'divide' matrices. | COEFFICIENT MATRIX: A matrix containing only the coefficients of the variables in a system of linear equations.

What's Next

What to Learn Next

Next, you can explore solving systems of three linear equations using matrices, which involves 3x3 matrices. This builds directly on what you've learned and is crucial for more advanced applications in science and engineering. Keep up the great work!