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.

2. How is the Determinant Calculated?

For a 2×2 matrix:

For larger matrices, the determinant is calculated using Laplace expansion (cofactor expansion):

• Choose any row or column
• For each element in that row/column, multiply it by (-1)^(i+j) and its minor
• Sum these products to get the determinant

3. Importance of Matrix Determinants

Applications: Determinants are used to solve systems of linear equations, find matrix inverses, determine if a matrix is invertible, and in calculating eigenvalues.

4. Using the Calculator

Format: Enter your matrix with comma-separated values within rows and semicolon-separated rows. Example: "1,2,3;4,5,6;7,8,9" for a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What matrices have determinants?
A: Only square matrices (n×n) have determinants. Rectangular matrices don't have determinants.

Q2: What does a zero determinant mean?
A: A zero determinant indicates the matrix is singular (not invertible) and the system of equations has either no solution or infinitely many solutions.

Q3: How does the determinant relate to area/volume?
A: The absolute value of the determinant gives the scaling factor of the linear transformation on area (2D) or volume (3D).

Q4: What's the computational complexity?
A: For an n×n matrix, the complexity is O(n!) for cofactor expansion, though more efficient methods exist.

Q5: Can determinants be negative?
A: Yes, determinants can be positive, negative, or zero, with the sign indicating orientation preservation.