Determinant
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📝 Step-by-step solution
Select matrix size and enter values.
📖 How to Use This Determinant Calculator
1
Select matrix size - Choose 2x2, 3x3, or 4x4 from the buttons above.
2
Enter matrix values - Fill in all the cells of the matrix with your numbers (decimals supported).
3
Click "Calculate Determinant" - Get the determinant value with step-by-step explanation.
📐 What is a Determinant?
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix:
- If det ≠ 0: The matrix is invertible (has an inverse)
- If det = 0: The matrix is singular (no inverse), rows/columns are linearly dependent
- The absolute value represents the scaling factor of the linear transformation
- The sign indicates whether the transformation preserves or reverses orientation
🧮 Determinant Formulas
2x2: det = ad - bc
3x3: det = a(ei - fh) - b(di - fg) + c(dh - eg)
4x4: Laplace expansion along any row or column
❓ Frequently Asked Questions
What does it mean if the determinant is zero?
If det(A) = 0, the matrix is singular meaning it has no inverse. The rows or columns are linearly dependent.
Can the determinant be negative?
Yes! A negative determinant indicates that the linear transformation includes a reflection (orientation reversal).
How do you calculate the determinant of a 4x4 matrix?
Use Laplace expansion along any row or column. Each element is multiplied by its cofactor (sign × minor determinant).
What is the difference between minor and cofactor?
The minor is the determinant of the submatrix after removing row i and column j. The cofactor = (-1)^(i+j) × minor.
What is the determinant of an identity matrix?
The determinant of an identity matrix I_n is always 1, regardless of size.