Pure & Applied MathematicsEntry № 03.11Documented Formula

Matrix Determinant Calculator (2×2 & 3×3)

Computes determinants for 2×2 and 3×3 square matrices with explicit Laplace cofactor expansions and geometric singularity checks.

Popular Searches:How to calculate 3x3 matrix determinant: a(ei-fh) - b(di-fg) + c(dh-eg)2x2 determinant formula: ad - bcSingular matrix determinant zero check

Laplace Matrix Determinant Calculator

Matrix Values

Expansion Steps:det = 1(1×0 − 4×6) − 2(0×0 − 4×5) + 3(0×6 − 1×5) = -24 − (-40) + (-15) = 1

Determinant det(A)

Invertibility: Invertible / Non-Singular (det ≠ 0)

Mathematical Proof & FormulaStandard Mathematical Notation

How This Calculation Formula is Formulated

Calculates the determinant scalar of square matrices via Laplace cofactor expansion, measuring the geometric scaling factor of linear transformations.

Standard Algebraic Representation

det(2×2) = ad − bc ; det(3×3) = a(ei − fh) − b(di − fg) + c(dh − eg)

Documented algebraic formula with transparent derivation and reference notes.

Variable Definitions & Measurement Units

Square Matrix

2×2 or 3×3 real matrix.

det(A)

Determinant

Scalar value. Non-zero indicates matrix is invertible.

Step-by-Step Calculation Example

Determinant of 3×3 Matrix

Computing det for row1 [1, 2, 3], row2 [0, 1, 4], row3 [5, 6, 0].

Cofactor Term 1

1 × (1×0 − 4×6) = 1 × (-24) = -24

First element

Cofactor Term 2

-2 × (0×0 − 4×5) = -2 × (-20) = +40

Second element (negated)

Cofactor Term 3

3 × (0×6 − 1×5) = 3 × (-5) = -15

Third element

Sum Terms

-24 + 40 − 15 = 1

det(A) = 1

Conclusion: det(A) = 1. Since det(A) ≠ 0, the matrix is invertible (non-singular).

In-Depth Editorial Analysis

Geometric Meaning of the Determinant

In 2D space, the absolute value of the determinant represents the area of the parallelogram formed by the column vectors. In 3D space, it represents the volume of the parallelepiped.

If det(A) = 0, the transformation collapses dimensions (squashing 3D space into a 2D plane or line), meaning the transformation has no inverse.

Frequently Asked Questions

Frequently Asked Questions About Matrix Determinant Calculator (2×2 & 3×3)

What does a negative determinant mean?

A negative determinant indicates that the linear transformation reverses the spatial orientation (e.g. reflects like a mirror).

Can a non-square matrix have a determinant?

No. Determinants are exclusively defined for square matrices (n × n).

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Deterministic Precision & 100% Client-Side Privacy

All calculations execute in your local browser using IEEE 754 double-precision floating-point arithmetic. Your figures and financial metrics remain private and are never uploaded or saved to any cloud servers.