How do you find the determinant of a 3×3 matrix?

Use cofactor expansion along the first row: det(A) = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁).

When does a matrix have an inverse?

A square matrix has an inverse if and only if its determinant is non-zero. Such a matrix is called 'non-singular' or 'invertible'.

How does matrix multiplication work?

For matrices A (m×n) and B (n×p), the product C = AB is m×p where each element C[i][j] = sum of A[i][k] × B[k][j] for k = 1 to n. The number of columns in A must equal the number of rows in B.

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Determinant 0

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Calculation Steps

Matrix Operations

Addition / Subtraction: Add or subtract corresponding elements. Both matrices must have the same dimensions.

Multiplication: C[i][j] = Σ A[i][k] × B[k][j]. The number of columns of A must equal the number of rows of B.

Determinant: A scalar value computed from a square matrix. For 2×2: ad − bc. For 3×3: cofactor expansion.

Transpose: Rows become columns: Aᵀ[i][j] = A[j][i].

Inverse: A⁻¹ exists only if det(A) ≠ 0. Computed as (1/det) × adj(A).

Determinant Formulas

2×2: det = a₁₁ × a₂₂ − a₁₂ × a₂₁
3×3 (Sarrus / Cofactor): Expand along the first row using cofactors.

How to Use

Select matrix size: 2×2 or 3×3.
Choose operation: Add, Subtract, Multiply, Determinant, Transpose, or Inverse.
Enter values: Fill in Matrix A (and Matrix B if needed).
View results: The result matrix, determinant, or inverse appears instantly with steps.

Properties

det(AB) = det(A) × det(B)
(AB)ᵀ = BᵀAᵀ
(A⁻¹)⁻¹ = A
det(A⁻¹) = 1 / det(A)

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