1. What is a Matrix Equation Solver?

Definition: This calculator solves for the vector y in the matrix equation Ay = b, where A is a square matrix and b is a vector.

Purpose: It helps students and professionals solve systems of linear equations represented in matrix form.

2. How Does the Calculator Work?

The calculator uses Gaussian elimination to solve:

Where:

• \( A \) — Coefficient matrix (n×n)
• \( \mathbf{y} \) — Unknown vector (n×1)
• \( \mathbf{b} \) — Constant terms vector (n×1)

Explanation: The system is transformed to row echelon form through elimination, then solved through back substitution.

3. Importance of Matrix Equation Solving

Details: Solving matrix equations is fundamental in engineering, physics, computer graphics, economics, and many other fields that model linear systems.

4. Using the Calculator

Tips:

• Enter matrix A with rows separated by semicolons and entries separated by commas (e.g., "1,2;3,4")
• Enter vector b with comma-separated values (e.g., "5,6")
• The system must be square (number of equations equals number of unknowns)

5. Frequently Asked Questions (FAQ)

Q1: What if my system has no solution?
A: The calculator will indicate if the system is inconsistent or cannot be solved.

Q2: Can I solve non-square systems?
A: No, this calculator only handles square systems (same number of equations as unknowns).

Q3: What precision does the calculator use?
A: Solutions are displayed with 4 decimal places, but calculations use full PHP floating-point precision.

Q4: How large can the matrix be?
A: For practical purposes, matrices up to about 10×10 will work well in this implementation.

Q5: What if my matrix is singular?
A: The calculator will detect this and indicate that the system cannot be solved uniquely.