Symbolab Matrix Determinant Calculator

Matrix Determinant Formula:

\[ \det(A) = \sum (-1)^{i+j} a_{ij} M_{ij} \text{ for expansion along row/column} \]

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 and the scaling factor of the transformation.

2. How Does the Calculator Work?

The calculator uses the Laplace expansion method:

\[ \det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij} \]

Where:

\( A \) — Square matrix
\( a_{ij} \) — Element in row i, column j
\( M_{ij} \) — Minor of element \( a_{ij} \)
\( (-1)^{i+j} \) — Sign factor

Explanation: The determinant is calculated by expanding along any row or column, multiplying each element by its cofactor (signed minor), and summing these products.

3. Importance of Determinant Calculation

Details: Determinants are crucial in linear algebra for solving systems of linear equations, finding inverses of matrices, determining linear independence, and in transformations in geometry.

4. Using the Calculator

Tips: Select the matrix size (2x2, 3x3, or 4x4) and enter all matrix elements. The calculator will compute the determinant using recursive expansion.

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: What's the fastest way to compute determinants?
A: For large matrices, LU decomposition is more efficient than Laplace expansion. For 2x2 and 3x3 matrices, direct formulas are fastest.

Q3: Can determinants be negative?
A: Yes, determinants can be positive, negative, or zero. The sign indicates whether the transformation preserves or reverses orientation.

Q4: What's the geometric meaning of the determinant?
A: The absolute value of the determinant gives the scaling factor of the linear transformation represented by the matrix.

Q5: Are there matrices without determinants?
A: Only square matrices have determinants. Non-square (rectangular) matrices do not have determinants.