1. What is an Augmented Matrix?

An augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. In linear algebra, an augmented matrix is used to represent a system of linear equations.

2. How Does the Calculator Work?

The calculator performs operations on augmented matrices:

Where:

• \( A \) — Coefficient matrix
• \( b \) — Right-hand side vector
• The vertical bar separates the coefficients from the constants

Operations: The calculator can transform the matrix to Reduced Row Echelon Form (RREF), solve the system, or calculate the matrix rank.

3. Importance of Augmented Matrices

Details: Augmented matrices are fundamental in solving systems of linear equations, determining linear independence, and understanding vector spaces. They provide a compact notation for performing Gaussian elimination.

4. Using the Calculator

Tips: Enter the dimensions of your matrix, fill in all elements (including the right-hand side after the vertical bar), and select the operation you want to perform.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between coefficient matrix and augmented matrix?
A: The coefficient matrix contains only the coefficients of the variables, while the augmented matrix includes an additional column for the constants.

Q2: What does RREF tell us about a system?
A: RREF reveals whether the system has no solution, exactly one solution, or infinitely many solutions.

Q3: How is rank related to solutions?
A: The rank determines the number of independent equations. If rank(A) = rank([A|b]), the system is consistent.

Q4: Can this calculator handle complex numbers?
A: This version handles real numbers only. Complex systems require specialized treatment.

Q5: What's the maximum size matrix this can handle?
A: The calculator is limited to 10×10 matrices for practical computation and display purposes.