Matrix Determinant Calculator

Matrix Determinant:

\[ \text{det}(A) = \sum (-1)^{i+j} a_{ij} M_{ij} \text{ for expansion along row/column, or for 2x2: } ad - bc \]

Determinant (det(A)):

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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.

2. How to Calculate Determinant

The calculator uses different methods based on matrix size:

For 2×2: \[ \text{det}\begin{pmatrix}a & b\\c & d\end{pmatrix} = ad - bc \] For 3×3: \[ \text{det}\begin{pmatrix}a & b & c\\d & e & f\\g & h & i\end{pmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Explanation: For larger matrices, the determinant is typically calculated using Laplace expansion (cofactor expansion) along a row or column.

3. Importance of Determinant

Details: The determinant is used in linear algebra to determine if a matrix is invertible, calculate the volume of a parallelepiped, solve systems of linear equations (Cramer's rule), and find eigenvalues.

4. Using the Calculator

Tips: Select matrix size (2x2 or 3x3), enter all matrix elements, and click Calculate. The calculator will compute the determinant using the appropriate method.

5. Frequently Asked Questions (FAQ)

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

Q2: Can I calculate determinant for non-square matrices?
A: No, determinants are only defined for square matrices (n×n).

Q3: What's the computational complexity of determinant calculation?
A: For an n×n matrix, the complexity is O(n!) for cofactor expansion, though more efficient methods exist.

Q4: How is determinant related to eigenvalues?
A: The determinant equals the product of all eigenvalues of the matrix.

Q5: What's the geometric interpretation of determinant?
A: The absolute value of the determinant gives the scaling factor of the linear transformation described by the matrix.