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 important properties of the matrix and the linear transformation it represents, such as whether the matrix is invertible and the scaling factor of the transformation.

2. How Does the Calculator Work?

The calculator uses the Laplace expansion (cofactor expansion) method:

Where:

• \( a_{ij} \) — Element of the matrix at row i, column j
• \( M_{ij} \) — Minor of the element at row i, column j
• The sum is taken along any row or column

Explanation: For each element in a row or column, multiply it by (-1)^(i+j) and by the determinant of the submatrix formed by deleting the current row and column.

3. Importance of Determinant Calculation

Details: Determinants are used in linear algebra for solving systems of linear equations, finding the inverse of a matrix, determining whether a matrix is singular, and in calculating eigenvalues.

4. Using the Calculator

Tips: Enter your square matrix using commas to separate elements in a row and semicolons to separate rows. For example, "1,2,3;4,5,6;7,8,9" represents a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What does the determinant tell us about a matrix?
A: The determinant indicates whether the matrix is invertible (non-zero determinant) and the scaling factor of the linear transformation represented by the matrix.

Q2: What's the geometric interpretation of determinant?
A: For 2×2 matrices, it's the area scaling factor; for 3×3 matrices, it's the volume scaling factor.

Q3: Can we calculate determinant for non-square matrices?
A: No, determinants are only defined for square matrices.

Q4: What's the time complexity for determinant calculation?
A: O(n!) for Laplace expansion, though more efficient methods like LU decomposition exist (O(n³)).

Q5: What does a zero determinant mean?
A: The matrix is singular (not invertible) and the system of equations has either no solution or infinitely many solutions.