What is an Augmented Matrix?

S6-SA1-0118

Grade Level:

Class 10

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

Definition

What is it?

An Augmented Matrix is a special type of matrix that combines the coefficients of a system of linear equations with their constant terms into a single matrix. It helps us solve these equations in a neat and organised way, making complex problems simpler.

Simple Example

Quick Example

Imagine you buy 2 samosas and 1 chai for Rs. 50, and your friend buys 1 samosa and 2 chai for Rs. 40. We can write this as two equations: 2S + 1C = 50 and 1S + 2C = 40. An augmented matrix would combine the numbers (2, 1, 50) and (1, 2, 40) into one big table.

Worked Example

Step-by-Step

Let's solve the system of equations:
x + 2y = 7
3x - y = 5

Step 1: Identify the coefficients of x and y, and the constant terms.
For the first equation: coefficients are 1 (for x), 2 (for y), and constant is 7.
For the second equation: coefficients are 3 (for x), -1 (for y), and constant is 5.
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Step 2: Write these coefficients in a matrix form. The left side is the coefficient matrix, and the right side is the constant matrix.
Coefficient matrix: [[1, 2], [3, -1]]
Constant matrix: [[7], [5]]
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Step 3: Combine them into an augmented matrix. We draw a vertical line (or just imagine it) to separate the coefficients from the constants.
[[1, 2 | 7],
[3, -1 | 5]]

Answer: The augmented matrix is [[1, 2 | 7], [3, -1 | 5]].

Why It Matters

Augmented matrices are super important for solving many equations quickly, which is key in fields like AI/ML for training models or in engineering to design structures. They help scientists in space technology calculate rocket trajectories and doctors in medicine understand complex biological systems.

Common Mistakes

MISTAKE: Forgetting to include a zero when a variable is missing in an equation. For example, if x + z = 5, writing only [1, 1, 5] | CORRECTION: Always include a zero for the missing variable. So, for x + z = 5 (meaning 1x + 0y + 1z = 5), the row should be [1, 0, 1 | 5].

MISTAKE: Mixing up the order of coefficients. For example, writing [2, 1 | 5] for 1x + 2y = 5 | CORRECTION: Ensure the coefficients for x, then y, then z (and so on) are always in the correct column order.

MISTAKE: Not placing the constant terms on the right side of the imaginary vertical line. | CORRECTION: The constant terms (the numbers on the right side of the equals sign) always form the last column of the augmented matrix.

Practice Questions

Try It Yourself

QUESTION: Convert the system of equations into an augmented matrix:
4x - y = 10
x + 5y = 2
| ANSWER: [[4, -1 | 10], [1, 5 | 2]]

QUESTION: Write the augmented matrix for:
a + 3b - c = 1
2a + 4c = 7
-b + 5c = 0
| ANSWER: [[1, 3, -1 | 1], [2, 0, 4 | 7], [0, -1, 5 | 0]]

QUESTION: If an augmented matrix is [[2, 0, 1 | 8], [0, 1, -3 | 5]], write the system of linear equations it represents. Assume variables are x, y, z. | ANSWER: 2x + z = 8, y - 3z = 5

MCQ

Quick Quiz

Which of the following represents the augmented matrix for the system: 5x + 2y = 11 and x - 3y = 4?

[[5, 2, 11], [1, -3, 4]]

[[5, 2 | 11], [1, -3 | 4]]

[[5, 1 | 11], [2, -3 | 4]]

[[11, 5, 2], [4, 1, -3]]

The Correct Answer Is:

Option B correctly places the coefficients of x and y on the left and the constant terms on the right, separated by the vertical line. The other options either mix up the order or don't use the correct augmented matrix format.

Real World Connection

In the Real World

Imagine a logistics company like Zepto or Dunzo planning delivery routes for multiple riders to deliver many orders. They use complex systems of equations to find the most efficient paths. Augmented matrices help computer programs quickly solve these equations, ensuring your chai reaches you hot and fast! This is a core part of operations research and supply chain management.

Key Vocabulary

Key Terms

MATRIX: A rectangular array of numbers or functions arranged in rows and columns | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression | LINEAR EQUATION: An equation between two variables that gives a straight line when plotted on a graph | CONSTANT TERM: A term in an algebraic expression that has a value that does not change (it has no variable attached)

What's Next

What to Learn Next

Great job understanding augmented matrices! Next, you should explore 'Gaussian Elimination' and 'Row Operations'. These are the powerful tools we use WITH augmented matrices to actually solve systems of equations and find the values of our variables.