4x4 Matrix Determinant Calculator

4x4 Matrix Determinant:

\[ \text{det}(A) = \begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{vmatrix} \]

Determinant:

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1. What is a 4x4 Matrix Determinant?

The determinant of a 4x4 matrix is a scalar value that can be computed from the elements of the matrix. It provides important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation it represents.

2. How Does the Calculator Work?

The calculator uses Laplace expansion (cofactor expansion) along the first row:

\[ \text{det}(A) = \sum_{j=1}^{4} (-1)^{1+j} a_{1j} \cdot \text{det}(M_{1j}) \]

Where:

\( a_{1j} \) — Elements of the first row
\( M_{1j} \) — The 3x3 minor matrix obtained by deleting the first row and j-th column
The determinant of each 3x3 minor is calculated using the standard rule of Sarrus

3. Importance of Matrix Determinant

Details: The determinant is fundamental in linear algebra. It's used to solve systems of linear equations, find inverse matrices, determine linear independence of vectors, and in various applications across physics and engineering.

4. Using the Calculator

Tips: Enter all 16 elements of your 4x4 matrix. The calculator will compute the determinant using cofactor expansion. For best results, enter exact values when possible.

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: Are there other methods to compute 4x4 determinants?
A: Yes, you can use row reduction to triangular form (where the determinant is the product of diagonal elements) or other expansion methods, but cofactor expansion is most straightforward for 4x4.

Q3: How accurate is this calculator?
A: The calculator performs exact arithmetic when possible, but floating-point precision limitations apply for non-integer values.

Q4: Can I use this for larger matrices?
A: This calculator is specifically for 4x4 matrices. For larger matrices, other methods like LU decomposition are more efficient.

Q5: What's the computational complexity?
A: For a 4x4 matrix using cofactor expansion, it requires calculating four 3x3 determinants, resulting in O(n!) time complexity for general n×n matrices.