1. What is a Matrix Determinant?

The determinant is a scalar value that can be computed from the elements of a square matrix. It encodes certain properties of the linear transformation described by the matrix, such as whether it's invertible and how it scales volumes.

2. How Does Expansion Along Row Work?

For a 4x4 matrix, the determinant can be calculated using expansion along any row:

Where:

• \( i \) — The row index (1-4)
• \( j \) — The column index (1-4)
• \( a_{ij} \) — The element at row i, column j
• \( M_{ij} \) — The 3x3 minor matrix obtained by removing row i and column j
• \( (-1)^{i+j} \) — The sign factor based on position

Explanation: This method recursively breaks down a 4x4 determinant into four 3x3 determinants, which are then calculated using standard formulas.

3. Importance of Determinant Calculation

Details: Determinants are fundamental in linear algebra, used to solve systems of linear equations, find matrix inverses, calculate eigenvalues, and determine whether a matrix is singular or invertible.

4. Using the Calculator

Tips: Enter all 16 values of your 4x4 matrix. The calculator will compute the determinant using expansion along the first row. For best results, use exact values when possible.

5. Frequently Asked Questions (FAQ)

Q1: What does a zero determinant indicate?
A: A determinant of zero means the matrix is singular (not invertible) and the system of equations it represents either has no solution or infinitely many solutions.

Q2: Can I use expansion along any row?
A: Yes, the expansion method works along any row or column, though choosing a row with zeros can simplify calculations.

Q3: Are there other methods to calculate determinants?
A: Yes, other methods include Gaussian elimination (row reduction), LU decomposition, and using eigenvalues.

Q4: What are the applications of determinants?
A: Determinants are used in calculus (Jacobian), physics (quantum mechanics), computer graphics (transformations), and economics (input-output models).

Q5: How accurate is this calculator?
A: The calculator provides results with 4 decimal places precision, but accuracy depends on the input values and potential floating-point arithmetic limitations.