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

S7-SA2-0248

Grade Level:

Class 12

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

Definition

What is it?

Solving a system of three linear equations using matrices means finding the unique values for three unknown variables (like x, y, and z) that satisfy all three equations at the same time. We represent these equations in a compact matrix form and use matrix operations to find the solution.

Simple Example

Quick Example

Imagine you go to a local kirana store. You buy 2 packets of biscuits, 1 bottle of juice, and 3 chocolates for Rs 120. Your friend buys 1 biscuit, 2 juices, and 1 chocolate for Rs 80. Another friend buys 3 biscuits, 1 juice, and 2 chocolates for Rs 150. If we want to find the price of each item (biscuit, juice, chocolate) using matrices, that's exactly what we're talking about!

Worked Example

Step-by-Step

Let's solve the system:
Equation 1: x + 2y - z = 3
Equation 2: 3x - y + 2z = 1
Equation 3: 2x + 3y + z = 7

Step 1: Write the system in matrix form AX = B.
A = [[1, 2, -1], [3, -1, 2], [2, 3, 1]]
X = [[x], [y], [z]]
B = [[3], [1], [7]]

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Step 2: Calculate the determinant of matrix A (det(A)).
det(A) = 1((-1)(1) - (2)(3)) - 2((3)(1) - (2)(2)) + (-1)((3)(3) - (-1)(2))
det(A) = 1(-1 - 6) - 2(3 - 4) - 1(9 + 2)
det(A) = 1(-7) - 2(-1) - 1(11)
det(A) = -7 + 2 - 11 = -16
Since det(A) is not zero, a unique solution exists.

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Step 3: Find the adjoint of matrix A (adj(A)).
First, find the cofactor matrix C.
C11 = -7, C12 = 1, C13 = 11
C21 = -5, C22 = 3, C23 = 1
C31 = 3, C32 = -5, C33 = -7
C = [[-7, 1, 11], [-5, 3, 1], [3, -5, -7]]
adj(A) = C_transpose = [[-7, -5, 3], [1, 3, -5], [11, 1, -7]]

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Step 4: Calculate the inverse of A (A_inverse = (1/det(A)) * adj(A)).
A_inverse = (1/-16) * [[-7, -5, 3], [1, 3, -5], [11, 1, -7]]

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Step 5: Solve for X using X = A_inverse * B.
X = (1/-16) * [[-7, -5, 3], [1, 3, -5], [11, 1, -7]] * [[3], [1], [7]]
X = (1/-16) * [[(-7)(3) + (-5)(1) + (3)(7)], [(1)(3) + (3)(1) + (-5)(7)], [(11)(3) + (1)(1) + (-7)(7)]]
X = (1/-16) * [[-21 - 5 + 21], [3 + 3 - 35], [33 + 1 - 49]]
X = (1/-16) * [[-5], [-29], [-15]]

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Step 6: Simplify to find x, y, and z.
x = -5/-16 = 5/16
y = -29/-16 = 29/16
z = -15/-16 = 15/16

Answer: x = 5/16, y = 29/16, z = 15/16

Why It Matters

Understanding how to solve systems of equations using matrices is super important! It's used in AI/ML to train models, in engineering to design bridges and circuits, and in finance to manage investments. Many real-world problems, from predicting weather to designing new medicines, rely on these powerful mathematical tools.

Common Mistakes

MISTAKE: Forgetting to check if the determinant of matrix A is zero. If it is zero, the inverse doesn't exist, and there might be no unique solution. | CORRECTION: Always calculate det(A) first. If det(A) = 0, state that the system either has no solution or infinitely many solutions, and the inverse method won't work.

MISTAKE: Errors in calculating the cofactors, especially getting the signs wrong (+/-) for each element. | CORRECTION: Remember the checkerboard pattern of signs: [[+, -, +], [-, +, -], [+, -, +]] when calculating cofactors. Double-check each calculation.

MISTAKE: Incorrectly multiplying matrices, especially forgetting the 'row by column' rule for multiplication. | CORRECTION: Practice matrix multiplication thoroughly. For A_inverse * B, ensure you multiply each row of A_inverse by the single column of B.

Practice Questions

Try It Yourself

QUESTION: Solve the system using matrices: x + y = 3, 2x - y = 0 | ANSWER: x = 1, y = 2

QUESTION: Find the values of x, y, z for: x - y + 2z = 1, 3x + y + z = 4, 2x - 3y + 4z = 2 | ANSWER: x = 1, y = 1, z = 0.5 (or 1/2)

QUESTION: For the system: 2x + y - z = 3, x - y + 2z = 1, 3x + 2y - 3z = 4, determine if a unique solution exists. If yes, find it. | ANSWER: det(A) = -8. A unique solution exists. x = 2, y = 1, z = 2

MCQ

Quick Quiz

Which condition must be true for a system of linear equations AX = B to have a unique solution using the matrix inverse method?

The determinant of A is zero.

The determinant of A is non-zero.

Matrix A is a 2x2 matrix.

Matrix B is a zero matrix.

The Correct Answer Is:

For a unique solution using the inverse method, the inverse of matrix A must exist, which means its determinant cannot be zero. If det(A) is zero, the inverse does not exist.

Real World Connection

In the Real World

Imagine ISRO scientists tracking three satellites in space. Each satellite's position can be described by equations. To find where they will meet or cross paths, or to adjust their trajectories, they use systems of linear equations and matrices. This helps them manage complex space missions and launch rockets accurately!

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 that tells us important properties, like if an inverse exists. | INVERSE MATRIX: A special matrix that, when multiplied by the original matrix, gives the identity matrix. | COFACTOR: A value used in calculating the determinant and inverse of a matrix. | ADJOINT MATRIX: The transpose of the cofactor matrix.

What's Next

What to Learn Next

Great job understanding this! Next, you should explore 'Cramer's Rule' for solving systems of linear equations. It's another powerful method that also uses determinants and is a good alternative to the inverse method, helping you solve problems even faster!