Matrix Determinant Calculator

About the Matrix Determinant Calculator

CalcPeak’s free Matrix Determinant Calculator simplifies the computation of determinants for square matrices up to 4x4, providing clear step-by-step solutions. Perfect for students learning linear algebra, teachers demonstrating matrix properties, or professionals in engineering, physics, and computer science, this tool uses the Laplace expansion method to compute determinants accurately for real number inputs (including negatives).

• What It Does: Computes the determinant of a square matrix, a scalar value essential for determining matrix invertibility, solving linear systems, and analyzing transformations. The calculator supports 2x2, 3x3, and 4x4 matrices, showing each step of the computation.
• Formulas:
  • 2x2 Matrix: For a matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the determinant is: \[ \det = ad - bc \]
  • 3x3 Matrix: For a matrix \( \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \), use Laplace expansion along the first row: \[ \det = a_{11} (-1)^{1+1} \det(M_{11}) + a_{12} (-1)^{1+2} \det(M_{12}) + a_{13} (-1)^{1+3} \det(M_{13}) \] where \( M_{ij} \) is the minor matrix obtained by removing row \( i \) and column \( j \).
  • 4x4 Matrix: Similarly, for a 4x4 matrix, expand along the first row: \[ \det = \sum_{j=1}^{4} a_{1j} (-1)^{1+j} \det(M_{1j}) \] This process repeats recursively for each 3x3 minor.
• Real-World Applications: Determinants are used in:
  • Linear Algebra: Checking matrix invertibility and solving systems via Cramer’s rule.
  • Geometry: Calculating area or volume scaling factors for transformations.
  • Physics: Analyzing electromagnetic fields or quantum mechanics.
  • Engineering: Assessing structural stability or circuit networks.
• How to Use:
  • Select the matrix size (2x2, 3x3, or 4x4) using the buttons.
  • Enter numbers in the matrix fields using the keypad (0–9, -, backspace).
  • Click "Calculate" to compute the determinant. Results and steps appear below.
  • Use "Clear" to reset inputs.
  • Copy results with the "Copy Results" button.
  • Share via WhatsApp, Facebook, or Twitter, or embed the calculator using the "Embed Calculator" button.
  • Explore related tools like our Linear Equation Solver for more linear algebra support.
• Example 1: 2x2 Matrix:
  • Input: \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
  • Steps:
    • Use the formula: \[ \det = ad - bc \]
    • Substitute: \[ \det = 1 \cdot 4 - 2 \cdot 3 \]
    • Compute: \[ = 4 - 6 = -2 \]
  • Result: \( \det = -2 \)
  • This shows the matrix is invertible (since \( \det \neq 0 \)).
• Example 2: 3x3 Matrix:
  • Input: \( \begin{bmatrix} 2 & -1 & 0 \\ 0 & 3 & -2 \\ 1 & 4 & 5 \end{bmatrix} \)
  • Steps:
    • Expand along the first row: \[ \det = 2 (-1)^{1+1} \det(M_{11}) + (-1) (-1)^{1+2} \det(M_{12}) + 0 (-1)^{1+3} \det(M_{13}) \]
    • Minor \( M_{11} \): \( \begin{bmatrix} 3 & -2 \\ 4 & 5 \end{bmatrix} \), \( \det = 3 \cdot 5 - (-2) \cdot 4 = 15 + 8 = 23 \)
    • Minor \( M_{12} \): \( \begin{bmatrix} 0 & -2 \\ 1 & 5 \end{bmatrix} \), \( \det = 0 \cdot 5 - (-2) \cdot 1 = 0 + 2 = 2 \)
    • Minor \( M_{13} \): Not needed since coefficient is 0.
    • Compute: \[ \det = 2 \cdot 23 + (-1) \cdot (-2) + 0 = 46 + 2 = 48 \]
  • Result: \( \det = 48 \)
  • This matrix is invertible and could represent a volume scaling factor in 3D geometry.
• Example 3: 4x4 Matrix:
  • Input: \( \begin{bmatrix} 1 & 0 & 2 & -1 \\ 3 & 2 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 2 & -1 & 0 & 1 \end{bmatrix} \)
  • Steps:
    • Expand along the first row: \[ \det = 1 (-1)^{1+1} \det(M_{11}) + 0 (-1)^{1+2} \det(M_{12}) + 2 (-1)^{1+3} \det(M_{13}) + (-1) (-1)^{1+4} \det(M_{14}) \]
    • Minor \( M_{11} \): \( \begin{bmatrix} 2 & 0 & 1 \\ 1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix} \), compute via 3x3 expansion (yields \( \det = -6 \)).
    • Minor \( M_{13} \): \( \begin{bmatrix} 3 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & -1 & 1 \end{bmatrix} \), \( \det = 3 \cdot (1 \cdot 1 - 0 \cdot (-1)) - 2 \cdot (0 \cdot 1 - 0 \cdot 2) + 1 \cdot (0 \cdot (-1) - 1 \cdot 2) = 3 \cdot 1 + 0 - 2 = 1 \).
    • Minor \( M_{14} \): \( \begin{bmatrix} 3 & 2 & 0 \\ 0 & 1 & 1 \\ 2 & -1 & 0 \end{bmatrix} \), \( \det = 3 \cdot (1 \cdot 0 - 1 \cdot (-1)) - 2 \cdot (0 \cdot 0 - 1 \cdot 2) + 0 = 3 \cdot 1 + 4 = 7 \).
    • Compute: \[ \det = 1 \cdot (-6) + 0 + 2 \cdot 1 + (-1) \cdot (-7) = -6 + 2 + 7 = 3 \]
  • Result: \( \det = 3 \)
  • This matrix is invertible and may represent a transformation in 4D space.
• Tips for Success:
  • Ensure all entries are valid numbers (negatives allowed).
  • Double-check inputs for 3x3 and 4x4 matrices to avoid errors in recursive calculations.
  • Use the backspace button to correct individual entries.
  • For more complex linear algebra tasks, try our Linear Equation Solver.

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