What is Solving Systems of Linear Equations using Matrix Inverse?

S7-SA2-0035

Grade Level:

Class 12

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

Definition

What is it?

Solving Systems of Linear Equations using Matrix Inverse is a powerful method to find the values of unknown variables in a set of linear equations. It converts the equations into a matrix form and then uses the inverse of a matrix to calculate the solution directly. This method is especially useful for systems with many equations.

Simple Example

Quick Example

Imagine you buy 2 samosas and 1 chai for Rs 50. Your friend buys 3 samosas and 2 chais for Rs 85. We can set up two equations: 2S + 1C = 50 and 3S + 2C = 85. Using matrices, we can quickly find the individual price of one samosa (S) and one chai (C).

Worked Example

Step-by-Step

Let's solve the system:
2x + 3y = 7
x + 2y = 4

---1. Write the equations in matrix form AX = B:
A = [[2, 3], [1, 2]] (Coefficient matrix)
X = [[x], [y]] (Variable matrix)
B = [[7], [4]] (Constant matrix)
So, [[2, 3], [1, 2]] * [[x], [y]] = [[7], [4]]

---2. Find the determinant of matrix A:
det(A) = (2 * 2) - (3 * 1) = 4 - 3 = 1

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

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

---5. Use the formula X = A^-1 * B to find X:
[[x], [y]] = [[2, -3], [-1, 2]] * [[7], [4]]

---6. Perform the matrix multiplication:
x = (2 * 7) + (-3 * 4) = 14 - 12 = 2
y = (-1 * 7) + (2 * 4) = -7 + 8 = 1

---Answer: So, x = 2 and y = 1.

Why It Matters

This method is super important in fields like AI/ML to train models, in Physics to solve complex circuit problems, and in Engineering to design stable structures. Future scientists and engineers use this to make self-driving cars safer, predict weather, and even create new medicines.

Common Mistakes

MISTAKE: Forgetting to find the determinant of the matrix A before calculating the inverse. | CORRECTION: Always calculate det(A) first. If det(A) is 0, the inverse does not exist, and the system either has no solution or infinitely many solutions.

MISTAKE: Incorrectly calculating the adjoint matrix by not swapping diagonal elements or not changing signs of off-diagonal elements. | CORRECTION: For a 2x2 matrix [[a,b],[c,d]], the adjoint is [[d,-b],[-c,a]]. Remember to swap 'a' and 'd', and negate 'b' and 'c'.

MISTAKE: Multiplying A^-1 and B in the wrong order (B * A^-1 instead of A^-1 * B). | CORRECTION: Matrix multiplication is not commutative. The correct order is always X = A^-1 * B.

Practice Questions

Try It Yourself

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

QUESTION: Find the values of a and b using matrix inverse for: 3a + 2b = 13, a + b = 5 | ANSWER: a = 3, b = 2

QUESTION: Given the system: 4x + y = 10 and 3x + 2y = 9. Find the inverse of the coefficient matrix and then use it to solve for x and y. | ANSWER: A^-1 = [[2/5, -1/5], [-3/5, 4/5]], x = 3, y = -2

MCQ

Quick Quiz

Which of the following is the correct formula to solve for the variable matrix X in AX = B using matrix inverse?

X = B * A^-1

X = A^-1 * B

X = A * B^-1

X = B^-1 * A

The Correct Answer Is:

The correct formula is X = A^-1 * B. We pre-multiply both sides of AX = B by A^-1 to isolate X, as matrix multiplication is not commutative.

Real World Connection

In the Real World

In India, companies like ISRO use this method to calculate satellite trajectories or manage complex sensor data. When you use navigation apps like Google Maps or Ola/Uber, the app's algorithm uses similar matrix calculations to figure out the shortest route or estimate arrival times based on multiple road conditions and traffic data.

Key Vocabulary

Key Terms

MATRIX: A rectangular arrangement of numbers or expressions in rows and columns. | INVERSE MATRIX: A special matrix that, when multiplied by the original matrix, results in an identity matrix. | DETERMINANT: A scalar value calculated from the elements of a square matrix. | COEFFICIENT MATRIX: A matrix containing the coefficients of the variables in a system of linear equations.

What's Next

What to Learn Next

Great job learning about matrix inverse! Next, you can explore Cramer's Rule, which is another method to solve systems of linear equations using determinants. It's a different approach but also relies on your understanding of matrices.