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Matrix Determinant Calculator

Matrix A

Square matrix of order 1 to 10. Rows and columns are synchronized. Use integers, decimals or fractions.

Determinant Result

det(A) = 0

Zero determinant: the matrix is singular and has no inverse.

Exact result using rational arithmetic.

Step by step

For A with rows (a, b) and (c, d), det(A) = ad − bc.

  1. Calculate the products ad and bc.
    0×0=0\mathbf{0 \times 0 = 0}
    0×0=0\mathbf{0 \times 0 = 0}
  2. Subtract the second total from the first.
    det⁡(A)=0−0=0\mathbf{\det(A) = 0 - 0 = 0}

How to Use the Determinant Calculator

Fill a single matrix A to calculate its determinant:

  1. Choose an order from 1 to 10 in Rows or Columns. Both fields stay synchronized.
  2. Confirm the size with Enter or by leaving the field, then fill each cell with an integer, decimal or fraction, such as 3/4.
  3. Read the updated result below the matrix or click Calculate determinant to confirm the entered size.

Empty or invalid entries are not silently converted to zero. Use zero or values with magnitude between 10⁻¹² and 10¹².

The original matrix remains unchanged. A zero determinant is a valid result and indicates a singular matrix.

Example: Determinant of a 2×2 Matrix

Consider matrix A with rows (4, 7) and (2, 6):

4
7
2
6

Multiply the main diagonal entries and subtract the product of the other diagonal: 4 × 6 − 7 × 2 = 10.

det⁡(A)=4×6−7×2=10\mathbf{\det(A) = 4 \times 6 - 7 \times 2 = 10}

The determinant is 10, so this matrix has an inverse. The two-diagonal rule above applies only to 2×2 matrices.

What Is a Matrix Determinant?

The determinant assigns a number to a square matrix. Among other properties, it indicates whether a linear transformation is invertible and its oriented area or volume scale factor.

The existence condition is having the same number of rows and columns. Rectangular matrices have no determinant under this definition.

Determinant Properties

  1. Identity: det(I) = 1.
  2. Row swap: swapping two rows changes the determinant's sign.
  3. Triangular matrix: the determinant is the product of the main diagonal entries.
  4. Product: det(AB) = det(A)det(B) for square matrices of the same order.
  5. Singularity: det(A) = 0 if and only if A has no inverse.

The determinant helps you study linear systems and transformations. When it is nonzero, you can also use the matrix inverse calculator.

Determinant Formulas and Method

For order 1, the determinant is the single entry. For order 2, use ad − bc. The calculator uses Gaussian elimination with partial pivoting for all supported orders.

det⁡([a])=a\mathbf{\det([a]) = a}
det⁡(abcd)=ad−bc\mathbf{\det\begin{pmatrix}a & b \\ c & d\end{pmatrix} = ad - bc}
det⁡(A)=(−1)s∏i=1nuii\mathbf{\det(A) = (-1)^s \prod_{i=1}^{n} u_{ii}}
nOrder of the square matrix.
sNumber of row swaps during elimination.
uᵢᵢDiagonal entries of the triangular matrix obtained without normalizing rows.

The method works on a copy of A. Results are approximate: cancellation between very close numbers can reduce precision.

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