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ALGEBRALinear AlgebraMathematics Calculator

adj

Adjoint (Adjugate) Matrix

The adjugate adj(A) is the transpose of the cofactor matrix. Key formula: A⁻¹ = adj(A) / det(A). A·adj(A) = det(A)·I.

Concept Fundamentals

adj(A)=(cof(A))ᵀ

Definition

adj(A)/det(A)

A⁻¹

A·adj(A)=det(A)·I

Property

[[d,-b],[-c,a]]

2×2

Compute AdjointTranspose of cofactor matrix

Why This Mathematical Concept Matters

Why: Adjoint gives a formula for A⁻¹ when det(A) ≠ 0. Used in Cramer's rule and theoretical derivations.

How: Compute cofactor matrix; transpose it. adj(A)[i,j] = Cⱼᵢ (note: Cⱼᵢ not Cᵢⱼ).

●adj(A) A = A adj(A) = det(A) I.
●adj(AB) = adj(B) adj(A).
●2×2: swap diagonal, negate off-diagonal.

Sources:Gilbert Strang, Linear AlgebraWolfram MathWorld

Quick Examples — Click to Load

Dimension

Matrix A (3×3)

adjoint_matrix.sh

CALCULATED

$ adjoint --size="3×3"

Size

3×3

Adjoint

computed

Original A (3×3)

0	0	0
0	0	0
0	0	0

Cofactor cof(A)

0	0	0
0	0	0
0	0	0

Adjoint adj(A) = (cof)ᵀ

0	0	0
0	0	0
0	0	0

Adjoint Matrix

3×3 adj(A)

[0, 0, 0] | [0, 0, 0] | [0, 0, 0]

numbervibe.com/calculators/mathematics/linear-algebra/adjoint-matrix-calculator

Adjoint Element Distribution (Bar)

Sign Distribution (Doughnut)

Calculation Steps

SETUP

Original A[0, 0, 0] [0, 0, 0] [0, 0, 0]

SETUP

Size3×3

FORMULA

Formulaadj(A) = (cof(A))ᵀ

INTERMEDIATE

Cofactor matrix[0, 0, 0] [0, 0, 0] [0, 0, 0]

RESULT

Transpose cof → adjSwap rows↔cols

RESULT

Adjoint adj(A)[0, 0, 0] [0, 0, 0] [0, 0, 0]

NOTE

RelationA⁻¹ = adj(A)/det(A), det=0

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

adj(adj(A)) = det(A)^(n−2) A

⚡

det(adj(A)) = det(A)^(n−1)

Key Takeaways

• Adjoint (adjugate) = transpose of cofactor matrix: adj(A) = (cof(A))ᵀ.
• Cofactor C_ij = (-1)^i+j × det(M_ij), M_ij = minor (submatrix).
• A⁻¹ = adj(A) / det(A) — inverse formula for non-singular A.
• A · adj(A) = adj(A) · A = det(A) · I.
• Singular matrices: adj still exists; A · adj(A) = 0 when det(A) = 0.

Did You Know?

📐2×2 shortcut: swap diagonal, negate off-diagonal.Source: Shortcut

🧮Cramer's rule uses adjoint for solving Ax = b.Source: Cramer

⚛️Cayley-Hamilton: adj appears in characteristic polynomial.Source: Theory

📊det(adj(A)) = det(A)^n-1 for n×n matrix.Source: Property

🔢adj(AB) = adj(B) · adj(A) — order reverses.Source: Product

🎯For singular A with rank n-1, adj has rank 1.Source: Rank

How It Works

For each element (i,j): find minor M_ij (determinant of submatrix with row i, col j removed), apply sign (-1)^i+j, build cofactor matrix, then transpose to get adjugate.

adj(A) = (cof(A))ᵀ

A⁻¹ = adj(A) / det(A)

Expert Tips

2×2 Shortcut

[[a,b],[c,d]] → adj = [[d,-b],[-c,a]].

Sign Pattern

Checkerboard: + − + − ... from (1,1).

Singular Case

adj exists even when det=0; A·adj(A)=0.

Inverse Formula

A⁻¹ = adj(A)/det(A) for invertible A.

Comparison Table

Feature	This Calculator	NumPy	Manual
Cofactor + Adjoint	✅	❌	⚠️
Bar & Doughnut charts	✅	❌	❌
8 preset examples	✅	❌	❌
Step-by-step	✅	❌	⚠️

FAQ

What is the adjoint matrix?

Transpose of the cofactor matrix. adj(A) = (cof(A))ᵀ.

How is it related to inverse?

A⁻¹ = adj(A) / det(A) for non-singular A.

Does adj exist for singular matrices?

Yes. A·adj(A)=0 when det(A)=0.

2×2 shortcut?

Swap diagonal elements, negate off-diagonal.

What is the cofactor?

C_ij = (-1)^(i+j) × det(minor_ij).

adj(AB) = ?

adj(B)·adj(A). Order reverses.

det(adj(A)) = ?

det(A)^(n-1) for n×n matrix.

When is adj symmetric?

When cofactor matrix is symmetric (e.g. symmetric A with special structure).

Stats

adj=(cof)ᵀ