Adjoint (Adjugate) Matrix Calculator (3x3)

What Is the Adjoint (Adjugate) Matrix?

The adjoint matrix, also called the adjugate matrix, is a foundational concept in linear algebra that enables exact computation of matrix inverses and provides the machinery behind Cramer's rule for solving linear systems. For a 3×3 matrix A, the adjoint is defined as the transpose of the cofactor matrix C, expressed as adj(A) = C^T. This identity appears as equation 148 in The Matrix Cookbook by Petersen and Pedersen and is the classical definition adopted by virtually every linear algebra textbook and computational library in use today. Unlike numerical decomposition methods, the adjugate formula produces exact integer or rational results from integer inputs, making it especially valuable in symbolic computation and cryptographic applications.

The Core Formula and Its Variables

Three nested definitions build the adjoint from the ground up:

• Minor M_ij: Delete row i and column j from A, then compute the determinant of the remaining 2×2 submatrix. Each of the nine minors of a 3×3 matrix involves exactly two multiplications and one subtraction.
• Cofactor C_ij: Attach a sign to the minor — C_ij = (−1)^i+j · M_ij. Positions where i+j is even keep a plus sign; positions where i+j is odd receive a minus sign.
• Adjoint adj(A): Transpose the full cofactor matrix so that adj(A)_ij = C_ji. Note the reversed subscript order — the entry at row i, column j of adj(A) comes from cofactor C_ji, not C_ij.

The alternating sign pattern for a 3×3 matrix forms the checkerboard grid: row 1 is [+, −, +], row 2 is [−, +, −], row 3 is [+, −, +].

Step-by-Step Calculation for a 3×3 Matrix

Step 1 — Label Each Entry

Denote the nine elements of A as a₁₁ through a₃₃, where the first subscript is the row index and the second is the column index. This calculator accepts entries a₁₁, a₁₂, a₁₃ (row 1), a₂₁, a₂₂, a₂₃ (row 2), and a₃₁, a₃₂, a₃₃ (row 3).

Step 2 — Compute All Nine 2×2 Minors

Delete one row and one column at a time and evaluate each resulting 2×2 determinant:

• M₁₁ = a₂₂a₃₃ − a₂₃a₃₂
• M₁₂ = a₂₁a₃₃ − a₂₃a₃₁
• M₁₃ = a₂₁a₃₂ − a₂₂a₃₁
• M₂₁ = a₁₂a₃₃ − a₁₃a₃₂
• M₂₂ = a₁₁a₃₃ − a₁₃a₃₁
• M₂₃ = a₁₁a₃₂ − a₁₂a₃₁
• M₃₁ = a₁₂a₂₃ − a₁₃a₂₂
• M₃₂ = a₁₁a₂₃ − a₁₃a₂₁
• M₃₃ = a₁₁a₂₂ − a₁₂a₂₁

Step 3 — Apply the Sign Pattern

Multiply each minor by (−1)^i+j. So C₁₁ = +M₁₁, C₁₂ = −M₁₂, C₁₃ = +M₁₃, C₂₁ = −M₂₁, C₂₂ = +M₂₂, and so on through C₃₃ = +M₃₃.

Step 4 — Transpose to Obtain the Adjugate

Swap the rows and columns of the cofactor matrix. The first column of adj(A) is taken from the first row of C; the second column from the second row; and so on. This transpose step is the most common source of manual calculation errors — always verify subscript placement after transposing.

Worked Numerical Example

Let A have entries: row 1 = [1, 2, 3], row 2 = [0, 4, 5], row 3 = [1, 0, 6]. Computing the nine minors and applying signs gives cofactors C₁₁ = 4·6 − 5·0 = 24, C₁₂ = −(0·6 − 5·1) = 5, C₁₃ = 0·0 − 4·1 = −4, C₂₁ = −(2·6 − 3·0) = −12, C₂₂ = 1·6 − 3·1 = 3, C₂₃ = −(1·0 − 2·1) = 2, C₃₁ = 2·5 − 3·4 = −2, C₃₂ = −(1·5 − 3·0) = −5, C₃₃ = 1·4 − 2·0 = 4. Transposing this cofactor matrix yields adj(A) with first row [24, −12, −2]. As established in Emory University's Section 3.2 on Determinants and Matrix Inverses, since det(A) = 22 for this matrix, the exact inverse is A⁻¹ = adj(A) / 22 — no row reduction required.

Key Applications

• Matrix Inversion: A⁻¹ = adj(A) / det(A) when det(A) ≠ 0 — the primary and defining use of the adjugate.
• Cramer's Rule: Express each variable in the system Ax = b as a ratio of determinants constructed from cofactors of A.
• 3D Computer Graphics: Transform surface normal vectors correctly under non-uniform scaling using the inverse-transpose, which derives directly from adj(A).
• Control Engineering: Compute resolvent matrices (sI − A)⁻¹ in state-space models for Laplace-domain stability analysis.
• Cryptography: Compute modular matrix inverses for Hill cipher encryption and decryption over integer residue fields.

Sources

• Petersen & Pedersen — The Matrix Cookbook, Equation 148
• Emory University — Section 3.2: Determinants and Matrix Inverses
• Cornell University — Lec 17: Inverse of a Matrix and Cramer's Rule
• Khan Academy — Inverting a 3×3 Matrix: Determinant and Adjugate