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Classical Adjoint — Definition, Formula & Examples

Adjugate
Classical Adjoint

The matrix formed by taking the transpose of the cofactor matrix of a given original matrix. The adjugate of matrix A is often written adj A.

Note: In the past, the term for adjugate used to be adjoint. The name has changed to avoid ambiguity with a different defintition of the term adjoint. Sometimes the adjugate is called the classical adjoint to refer to this older term. Consult a book on linear algebra for more information.

 

Example:

Find the adjugate of the following matrix:

\[{\rm{A}} = \left[ {\begin{array}{*{20}c} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \\\end{array}} \right]\]
Solution:

First find the cofactor of each element.

\[\begin{array}{*{20}{l}}{{{\rm{A}}_{11}} = \left| {\begin{array}{*{20}{c}}4&5\\0&6\end{array}} \right| = 24}&{{{\rm{A}}_{12}} = - \left| {\begin{array}{*{20}{c}}0&5\\1&6\end{array}} \right| = 5}&{{{\rm{A}}_{13}} = \left| {\begin{array}{*{20}{c}}0&4\\1&0\end{array}} \right| = - 4}\\{}&{}&{}\\{{{\rm{A}}_{21}} = - \left| {\begin{array}{*{20}{c}}2&3\\0&6\end{array}} \right| = - 12}&{{{\rm{A}}_{22}} = \left| {\begin{array}{*{20}{c}}1&3\\1&6\end{array}} \right| = 3}&{{{\rm{A}}_{23}} = - \left| {\begin{array}{*{20}{c}}1&2\\1&0\end{array}} \right| = 2}\\{}&{}&{}\\{{{\rm{A}}_{31}} = \left| {\begin{array}{*{20}{c}}2&3\\4&5\end{array}} \right| = - 2}&{{{\rm{A}}_{32}} = - \left| {\begin{array}{*{20}{c}}1&3\\0&5\end{array}} \right| = - 5}&{{{\rm{A}}_{33}} = \left| {\begin{array}{*{20}{c}}1&2\\0&4\end{array}} \right| = 4}\end{array}\]

As a result the cofactor matrix of A is

\[\left[ {\begin{array}{*{20}{c}}{24}&5&{ - 4}\\{ - 12}&3&2\\{ - 2}&{ - 5}&4\end{array}} \right]\]

Finally the adjugate of A is the transpose of the cofactor matrix:

\[\left[ {\begin{array}{*{20}{c}}{24}&{ - 12}&{ - 2}\\5&3&{ - 5}\\{ - 4}&2&4\end{array}} \right]\]

 

See also

Inverse of a matrix

Key Formula

A−1=1det⁡(A)⋅adj⁡(A)A^{-1} = \frac{1}{\det(A)} \cdot \operatorname{adj}(A)
Where:
  • AA = An n × n square matrix that is invertible
  • adj⁡(A)\operatorname{adj}(A) = The classical adjoint (adjugate) of A, equal to the transpose of the cofactor matrix of A
  • det⁡(A)\det(A) = The determinant of A, which must be nonzero for the inverse to exist
  • A−1A^{-1} = The inverse of matrix A

Worked Example

Problem: Find the classical adjoint of the 2×2 matrix B and use it to compute B⁻¹, where B = [[3, 1], [2, 4]].
Step 1: Find all four cofactors. For a 2×2 matrix, each cofactor is the signed minor of the corresponding element.
C11=+4,C12=−2,C21=−1,C22=+3C_{11} = +4, \quad C_{12} = -2, \quad C_{21} = -1, \quad C_{22} = +3
Step 2: Write the cofactor matrix by placing each cofactor in its corresponding position.
Cofactor matrix=[4−2−13]\text{Cofactor matrix} = \begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix}
Step 3: Transpose the cofactor matrix to get the classical adjoint (adjugate).
adj⁡(B)=[4−1−23]\operatorname{adj}(B) = \begin{bmatrix} 4 & -1 \\ -2 & 3 \end{bmatrix}
Step 4: Compute the determinant of B.
det⁡(B)=(3)(4)−(1)(2)=10\det(B) = (3)(4) - (1)(2) = 10
Step 5: Divide the adjugate by the determinant to find the inverse.
B−1=110[4−1−23]=[0.4−0.1−0.20.3]B^{-1} = \frac{1}{10}\begin{bmatrix} 4 & -1 \\ -2 & 3 \end{bmatrix} = \begin{bmatrix} 0.4 & -0.1 \\ -0.2 & 0.3 \end{bmatrix}
Answer: The classical adjoint is adj(B) = [[4, −1], [−2, 3]], and the inverse is B⁻¹ = (1/10)·[[4, −1], [−2, 3]].

Another Example

This 3×3 example demonstrates the important identity A · adj(A) = det(A) · I, which is the theoretical basis for why the adjugate method works for computing inverses.

Problem: Find the classical adjoint of the 3×3 matrix C = [[2, 0, 1], [0, 1, 0], [1, 0, 2]] and verify that C · adj(C) = det(C) · I.
Step 1: Compute each cofactor. The signs follow the checkerboard pattern (+, −, +, −, …).
C11=∣1002∣=2,C12=−∣0012∣=0,C13=∣0110∣=−1C_{11} = \begin{vmatrix}1&0\\0&2\end{vmatrix} = 2, \quad C_{12} = -\begin{vmatrix}0&0\\1&2\end{vmatrix} = 0, \quad C_{13} = \begin{vmatrix}0&1\\1&0\end{vmatrix} = -1
Step 2: Continue with the second and third rows.
C21=−∣0102∣=0,C22=∣2112∣=3,C23=−∣2010∣=0C_{21} = -\begin{vmatrix}0&1\\0&2\end{vmatrix} = 0, \quad C_{22} = \begin{vmatrix}2&1\\1&2\end{vmatrix} = 3, \quad C_{23} = -\begin{vmatrix}2&0\\1&0\end{vmatrix} = 0
Step 3: Third row cofactors:
C31=∣0110∣=−1,C32=−∣2100∣=0,C33=∣2001∣=2C_{31} = \begin{vmatrix}0&1\\1&0\end{vmatrix} = -1, \quad C_{32} = -\begin{vmatrix}2&1\\0&0\end{vmatrix} = 0, \quad C_{33} = \begin{vmatrix}2&0\\0&1\end{vmatrix} = 2
Step 4: Transpose the cofactor matrix to obtain the classical adjoint.
adj⁡(C)=[20−1030−102]\operatorname{adj}(C) = \begin{bmatrix}2&0&-1\\0&3&0\\-1&0&2\end{bmatrix}
Step 5: Verify: compute det(C) = 2(2) − 0 + 1(−1) = 3, then multiply C · adj(C).
C⋅adj⁡(C)=[300030003]=3I  ✓C \cdot \operatorname{adj}(C) = \begin{bmatrix}3&0&0\\0&3&0\\0&0&3\end{bmatrix} = 3I \; \checkmark
Answer: adj(C) = [[2, 0, −1], [0, 3, 0], [−1, 0, 2]], and the verification confirms C · adj(C) = 3I = det(C) · I.

Frequently Asked Questions

What is the difference between the classical adjoint and the adjoint of a matrix?
The classical adjoint (adjugate) is the transpose of the cofactor matrix. The term 'adjoint' in modern linear algebra usually refers to the conjugate transpose (also called the Hermitian adjoint), denoted A*. The two concepts are entirely different; 'classical adjoint' is used specifically to avoid this confusion.
How do you find the classical adjoint of a 2×2 matrix quickly?
For a 2×2 matrix [[a, b], [c, d]], the classical adjoint is simply [[d, −b], [−c, a]]. You swap the diagonal entries and negate the off-diagonal entries. There is no need to compute individual cofactors separately.
When do you use the classical adjoint to find an inverse?
You can always use adj(A) to find A⁻¹ via the formula A⁻¹ = (1/det A) · adj(A), provided det(A) ≠ 0. For 2×2 matrices, this is the fastest hand-calculation method. For larger matrices, row reduction is often more efficient, but the adjugate formula is important in theoretical proofs and for deriving Cramer's rule.

Classical Adjoint (Adjugate) vs. Adjoint (Conjugate Transpose)

Classical Adjoint (Adjugate)Adjoint (Conjugate Transpose)
DefinitionTranspose of the cofactor matrixTranspose with complex conjugation of each entry
Notationadj(A)A* or A†
Applies toSquare matrices (real or complex)Any matrix (real or complex), especially in inner product spaces
Primary useComputing matrix inverse via A⁻¹ = adj(A)/det(A)Defining self-adjoint (Hermitian) matrices, quantum mechanics, functional analysis
For real matricesAlways different from the ordinary transposeReduces to the ordinary transpose

Why It Matters

The classical adjoint appears directly in the formula for the inverse of a matrix, which is one of the most important results in linear algebra. It also provides the theoretical foundation for Cramer's rule, where each component of the solution vector involves cofactors from the adjugate. Students encounter this concept in precalculus and college-level linear algebra courses whenever they need to invert matrices by hand or prove properties of determinants.

Common Mistakes

Mistake: Forgetting to transpose the cofactor matrix.
Correction: The classical adjoint is NOT the cofactor matrix itself — it is the transpose of the cofactor matrix. After computing all cofactors, you must swap rows and columns before calling the result adj(A).
Mistake: Using incorrect signs on the cofactors.
Correction: Each cofactor carries a sign factor of (−1)^(i+j). A common error is applying the wrong sign or forgetting it altogether. Use the checkerboard pattern: + − + / − + − / + − + for a 3×3 matrix.

Related Terms

  • Cofactor Matrix — The matrix you transpose to get the adjugate
  • Cofactor — Signed minor of each matrix entry
  • Transpose of a Matrix — Operation applied to cofactor matrix to form adjugate
  • Inverse of a Matrix — Computed using adj(A) divided by det(A)
  • Matrix — The fundamental object that the adjugate operates on
  • Determinant — Scalar factor used alongside adjugate for inverses
  • Minor — Sub-determinant used to compute each cofactor