Matrix Determinant Calculator
Matrix Determinant Calculator computes the determinant of a square matrix (up to 3×3) with detailed step-by-step Laplace expansion, ideal for linear algebra.
Formulas Used
The determinant of a matrix is a scalar value that provides important properties about the matrix, such as invertibility.
- 2×2 Matrix:
For a matrix \\( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \\):
\\[ \det(A) = ad – bc \\]
Where:
- \\( a, b, c, d \\): Matrix elements
- 3×3 Matrix:
For a matrix \\( \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \\):
\\[ \det(A) = a(ei – fh) – b(di – fg) + c(dh – eg) \\]
Where:
- \\( a, b, c, d, e, f, g, h, i \\): Matrix elements
Example Calculations
Example 1: 2×2 Matrix
Input: Matrix = \\( \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} \\)
Calculations:
- Determinant: \\[ \det(A) = (4 \cdot 1) – (3 \cdot 2) = 4 – 6 = -2 \\]
Result: Determinant = -2
Example 2: 3×3 Matrix
Input: Matrix = \\( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \\)
Calculations:
- Determinant: \\[ \det(A) = 1(5 \cdot 9 – 6 \cdot 8) – 2(4 \cdot 9 – 6 \cdot 7) + 3(4 \cdot 8 – 5 \cdot 7) \\]
- \\[ = 1(45 – 48) – 2(36 – 42) + 3(32 – 35) \\]
- \\[ = 1(-3) – 2(-6) + 3(-3) = -3 + 12 – 9 = 0 \\]
Result: Determinant = 0
Example 3: 2×2 Matrix
Input: Matrix = \\( \begin{bmatrix} 0 & -1 \\ 2 & 3 \end{bmatrix} \\)
Calculations:
- Determinant: \\[ \det(A) = (0 \cdot 3) – (-1 \cdot 2) = 0 + 2 = 2 \\]
Result: Determinant = 2
How to Use the Calculator
Follow these steps to calculate the determinant of a matrix:
- Select Matrix Size: Choose 2×2 or 3×3 from the dropdown.
- Enter Matrix Elements: Input integer values for each matrix element (positive or negative).
- Calculate: Click “Calculate Determinant” to see the result.
- Interpret Result: The result shows the determinant value with detailed calculations.
- Share or Embed: Use the share buttons to post results on social media, copy the result, or get an embed code.
Note: All inputs must be valid integers. Invalid or missing inputs will prompt an error. The determinant is calculated using standard formulas for 2×2 and 3×3 matrices.