What is Solving Linear Equations using Matrices?

S6-SA1-0201

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Solving Linear Equations using Matrices is a powerful method to find the values of unknown variables in a system of linear equations. Instead of traditional substitution or elimination, we represent the equations as matrices and use matrix operations to find the solution. It's like organizing your equations into neat rows and columns to solve them more efficiently.

Simple Example

Quick Example

Imagine you bought 2 samosas and 1 chai for Rs 50. Your friend bought 3 samosas and 2 chais for Rs 90. If we want to find the price of one samosa and one chai, we can write these as equations. Using matrices helps us quickly find the individual prices without guessing or trying multiple combinations.

Worked Example

Step-by-Step

Let's solve the system:
2x + 3y = 12
x + 2y = 7

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Step 1: Write the equations in matrix form AX = B.
A = [[2, 3], [1, 2]] (Coefficient Matrix)
X = [[x], [y]] (Variable Matrix)
B = [[12], [7]] (Constant Matrix)

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Step 2: Find the determinant of matrix A (det(A)).
det(A) = (2 * 2) - (3 * 1) = 4 - 3 = 1
Since det(A) is not 0, a unique solution exists.

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

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

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Step 5: Calculate X = A^-1 * B.
X = [[2, -3], [-1, 2]] * [[12], [7]]

X = [[(2 * 12) + (-3 * 7)], [(-1 * 12) + (2 * 7)]]
X = [[24 - 21], [-12 + 14]]
X = [[3], [2]]

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Step 6: Identify the values of x and y.
x = 3, y = 2

Answer: x = 3, y = 2

Why It Matters

This method is super important in fields like AI/ML for training models, in Physics for solving complex circuit problems, and in Engineering for designing structures. Scientists at ISRO use it to calculate rocket trajectories, and doctors use it in medical imaging. Mastering this opens doors to exciting careers in technology and research!

Common Mistakes

MISTAKE: Forgetting to find the inverse of the coefficient matrix | CORRECTION: Remember that to solve AX=B, you need to calculate X = A^-1 * B. Finding A^-1 is a crucial step.

MISTAKE: Incorrectly calculating the determinant or adjoint of the matrix | CORRECTION: Double-check your determinant calculation (ad - bc for a 2x2 matrix) and ensure you swap diagonal elements and change signs for off-diagonal elements correctly when finding the adjoint.

MISTAKE: Mixing up the order of matrix multiplication (e.g., B * A^-1 instead of A^-1 * B) | CORRECTION: Matrix multiplication is not commutative. Always multiply A^-1 by B from the left side, i.e., X = A^-1 * B.

Practice Questions

Try It Yourself

QUESTION: Solve using matrices: x + y = 5, 2x - y = 1 | ANSWER: x = 2, y = 3

QUESTION: Solve using matrices: 3x - 2y = 7, x + 4y = 9 | ANSWER: x = 3, y = 1

QUESTION: If the system 4x + 2y = 10 and 2x + y = k has no unique solution, what is the value of k? | ANSWER: This system will have no unique solution if the determinant of the coefficient matrix is 0. The determinant is (4*1 - 2*2) = 0. This means the lines are parallel or coincident. For no unique solution, the lines must be parallel and distinct. In this case, for no unique solution, it means the equations are multiples of each other but with different constants, or parallel. If the determinant is 0, it indicates either infinite solutions or no solution. The question asks for 'no unique solution', which covers both. No specific k value is required for the determinant to be 0, as it's already 0.

MCQ

Quick Quiz

Which of the following is the correct matrix equation form for solving a system of linear equations?

X = AB

A = BX

AX = B

B = XA

The Correct Answer Is:

The standard way to represent a system of linear equations in matrix form is AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Real World Connection

In the Real World

When you use GPS on your phone to find the shortest route, the app is solving many linear equations using matrices to calculate distances and optimal paths. Similarly, in cricket analytics, matrices help process vast amounts of data to predict player performance or team strategies during an IPL match. Even in managing traffic signals in a busy Indian city, matrices help optimize signal timings to reduce congestion.

Key Vocabulary

Key Terms

MATRIX: A rectangular arrangement of numbers or functions in rows and columns | DETERMINANT: A scalar value calculated from the elements of a square matrix, indicating properties like invertibility | INVERSE MATRIX: A matrix that, when multiplied by the original matrix, yields the identity matrix | COEFFICIENT MATRIX: A matrix containing the coefficients of the variables in a system of linear equations | ADJOINT MATRIX: The transpose of the cofactor matrix of a given square matrix.

What's Next

What to Learn Next

Great job understanding this! Next, you can explore 'Solving Systems of Linear Equations using Cramer's Rule'. It's another method that builds on determinants and provides an alternative way to solve these equations, giving you more tools for problem-solving.