Determinant Calculator

Determinant is evaluated from Matrix Size, Row 1, Col 1 and Row 1, Col 2. The calculation reports Determinant |A|, Trace tr and Singular?.

Results

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About the Determinant Calculator

Determinant is treated here as a quantitative relation between Matrix Size, Row 1, Col 1, Row 1, Col 2 and Row 1, Col 3 and Determinant |A|, Trace tr, Singular? and A⁻¹ [1,1].

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
2 x 2: det = ad - bc
Inverse: A⁻¹ = (1/det) x adjugate(A)

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

2 x 2: det = ad - bc
Inverse: A⁻¹ = (1/det) x adjugate(A)

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: 2×2 Determinant — Basic

Inputs

size: 2×2 a11: 3 a12: 2 a21: 1 a22: 4
Determinant |A|: 10. Trace tr: 7. Singular?: No - matrix is invertible. A⁻¹ [1,1]: 0.4. A⁻¹ [1,2]: -0.2. A⁻¹ [2,1]: -0.1. A⁻¹ [2,2]: 0.3. Area of Parallelogram |det|: 10 units^2

With Matrix Size = 2 x 2, Row 1, Col 1 = 3, Row 1, Col 2 = 2 and Row 2, Col 1 = 1 as the stated inputs, the result is Determinant |A| = 10, Trace tr = 7 and Singular? = No - matrix is invertible. Each value corresponds to the declared output fields.

Example 2: Singular Matrix — No Inverse

Inputs

size: 2×2 a11: 2 a12: 4 a21: 1 a22: 2
Determinant |A|: 0. Trace tr: 4. Singular?: Yes - matrix is singular (no inverse). A⁻¹ [1,1]: 0. A⁻¹ [1,2]: 0. A⁻¹ [2,1]: 0. A⁻¹ [2,2]: 0. Area of Parallelogram |det|: 0 units^2

With Matrix Size = 2 x 2, Row 1, Col 1 = 2, Row 1, Col 2 = 4 and Row 2, Col 1 = 1 as the stated inputs, the result is Determinant |A| = 0, Trace tr = 4 and Singular? = Yes - matrix is singular (no inverse). Each value corresponds to the declared output fields.

Example 3: 3×3 Determinant — Volume

Inputs

size: 3×3 a11: 1 a12: 0 a13: 0 a21: 0 a22: 2 a23: 0 a31: 0 a32: 0 a33: 3
Determinant |A|: 6. Trace tr: 6. Singular?: No - matrix is invertible. A⁻¹ [1,1]: 0. A⁻¹ [1,2]: 0. A⁻¹ [2,1]: 0. A⁻¹ [2,2]: 0. Area of Parallelogram |det|: 6 units^2

With Matrix Size = 3 x 3, Row 1, Col 1 = 1, Row 1, Col 2 = 0 and Row 1, Col 3 = 0 as the stated inputs, the result is Determinant |A| = 6, Trace tr = 6 and Singular? = No - matrix is invertible. Each value corresponds to the declared output fields.

Example 4: Cramer's Rule — Solve 2×2 System

Inputs

size: 2×2 a11: 2 a12: 1 a21: 5 a22: 3
Determinant |A|: 1. Trace tr: 5. Singular?: No - matrix is invertible. A⁻¹ [1,1]: 3. A⁻¹ [1,2]: -1. A⁻¹ [2,1]: -5. A⁻¹ [2,2]: 2. Area of Parallelogram |det|: 1 units^2

With Matrix Size = 2 x 2, Row 1, Col 1 = 2, Row 1, Col 2 = 1 and Row 2, Col 1 = 5 as the stated inputs, the result is Determinant |A| = 1, Trace tr = 5 and Singular? = No - matrix is invertible. Each value corresponds to the declared output fields.

Common Use Cases

  • Calculate 2×2 determinant for area of parallelogram
  • Find matrix inverse for solving linear systems
  • Use Cramer's rule for system of equations
  • Check if matrix is singular (det=0)