Adjugate matrix

In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix.

The adjugate has sometimes been called the "adjoint", but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.

Definition

The adjugate of A is the transpose of the cofactor matrix C of A,

$\mathrm {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}~.$$\mathrm{adj}(\mathbf{A}) = \mathbf{C}^\mathsf{T} ~.$

In more detail, suppose R is a commutative ring and A is an n×n matrix with entries from R.

• The (i,j) minor of A, denoted M_ij, is the determinant of the (n − 1)×(n − 1) matrix that results from deleting row i and column j of A.
• The cofactor matrix of A is the n×n matrix C whose (i, j) entry is the (i, j) cofactor of A,

$\mathbf {C} _{ij}=(-1)^{i+j}\mathbf {M} _{ij}~.$$\mathbf {C} _{ij}=(-1)^{i+j}\mathbf {M} _{ij}~.$

• The adjugate of A is the transpose of C, that is, the n×n matrix whose (i,j) entry is the (j,i) cofactor of A,

$\mathrm {adj} (\mathbf {A} )_{ij}=\mathbf {C} _{ji}=(-1)^{i+j}\mathbf {M} _{ji}\,$${\mathrm {adj}}({\mathbf {A}})_{{ij}}={\mathbf {C}}_{{ji}}=(-1)^{{i+j}}{\mathbf {M}}_{{ji}}\,$.

The adjugate is defined as it is so that the product of A with its adjugate yields a diagonal matrix whose diagonal entries are det(A),

$\mathbf {A} \,\mathrm {adj} (\mathbf {A} )=\det(\mathbf {A} )\,\mathbf {I} ~.$$\mathbf{A} \, \mathrm{adj}(\mathbf{A}) = \det(\mathbf{A}) \, \mathbf{I} ~.$

A is invertible if and only if det(A) is an invertible element of R, and in that case the equation above yields

$\mathrm {adj} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {A} ^{-1}~,$$\mathrm{adj}(\mathbf{A}) = \det(\mathbf{A}) \mathbf{A}^{-1} ~,$

$\mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\,\mathrm {adj} (\mathbf {A} )~.$$\mathbf{A}^{-1} = \frac {1} {\det(\mathbf{A})} \, \mathrm{adj}(\mathbf{A}) ~.$

Examples

2 × 2 generic matrix

The adjugate of the 2 × 2 matrix

$\mathbf {A} ={\begin{pmatrix}{a}&{b}\\{c}&{d}\end{pmatrix}}$${\mathbf {A}}={\begin{pmatrix}{{a}}&{{b}}\\{{c}}&{{d}}\end{pmatrix}}$

is

$\operatorname {adj} (\mathbf {A} )={\begin{pmatrix}\,\,\,{d}&\!\!{-b}\\{-c}&{a}\end{pmatrix}}$$\operatorname {adj}({\mathbf {A}})={\begin{pmatrix}\,\,\,{{d}}&\!\!{{-b}}\\{{-c}}&{{a}}\end{pmatrix}}$.

It is seen that det(adj(A)) = det(A) and hence that adj(adj(A)) = A.

3 × 3 generic matrix

Consider a 3×3 matrix

$\mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}$$\mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}$

Its cofactor matrix is

$\mathbf {C} ={\begin{pmatrix}+{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}}\\&&\\-{\begin{vmatrix}a_{12}&a_{13}\\a_{32}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{13}\\a_{31}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{12}\\a_{31}&a_{32}\end{vmatrix}}\\&&\\+{\begin{vmatrix}a_{12}&a_{13}\\a_{22}&a_{23}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{13}\\a_{21}&a_{23}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}\end{pmatrix}}$$\mathbf {C} ={\begin{pmatrix}+{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}}\\&&\\-{\begin{vmatrix}a_{12}&a_{13}\\a_{32}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{13}\\a_{31}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{12}\\a_{31}&a_{32}\end{vmatrix}}\\&&\\+{\begin{vmatrix}a_{12}&a_{13}\\a_{22}&a_{23}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{13}\\a_{21}&a_{23}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}\end{pmatrix}}$

where ${\begin{vmatrix}a_{im}&a_{in}\\a_{jm}&a_{jn}\end{vmatrix}}=\det {\begin{pmatrix}a_{im}&a_{in}\\a_{jm}&a_{jn}\end{pmatrix}}$${\begin{vmatrix}a_{im}&a_{in}\\a_{jm}&a_{jn}\end{vmatrix}}=\det {\begin{pmatrix}a_{im}&a_{in}\\a_{jm}&a_{jn}\end{pmatrix}}$.

Its adjugate is the transpose of its cofactor matrix:

$\operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}={\begin{pmatrix}+{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{12}&a_{13}\\a_{32}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{12}&a_{13}\\a_{22}&a_{23}\end{vmatrix}}\\&&\\-{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{13}\\a_{31}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{13}\\a_{21}&a_{23}\end{vmatrix}}\\&&\\+{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{12}\\a_{31}&a_{32}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}\end{pmatrix}}$$\operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}={\begin{pmatrix}+{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{12}&a_{13}\\a_{32}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{12}&a_{13}\\a_{22}&a_{23}\end{vmatrix}}\\&&\\-{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{13}\\a_{31}&a_{33}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{13}\\a_{21}&a_{23}\end{vmatrix}}\\&&\\+{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}}&-{\begin{vmatrix}a_{11}&a_{12}\\a_{31}&a_{32}\end{vmatrix}}&+{\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}}\end{pmatrix}}$

3 × 3 numeric matrix

As a specific example, we have

$\operatorname {adj} {\begin{pmatrix}\!-3&\,2&\!-5\\\!-1&\,0&\!-2\\\,3&\!-4&\,1\end{pmatrix}}={\begin{pmatrix}\!-8&\,18&\!-4\\\!-5&\!12&\,-1\\\,4&\!-6&\,2\end{pmatrix}}$$\operatorname {adj}{\begin{pmatrix}\!-3&\,2&\!-5\\\!-1&\,0&\!-2\\\,3&\!-4&\,1\end{pmatrix}}={\begin{pmatrix}\!-8&\,18&\!-4\\\!-5&\!12&\,-1\\\,4&\!-6&\,2\end{pmatrix}}$.

The −6 in the third row, second column of the adjugate was computed as follows:

$(-1)^{2+3}\;\operatorname {det} {\begin{pmatrix}\!-3&\,2\\\,3&\!-4\end{pmatrix}}=-((-3)(-4)-(3)(2))=-6.$$(-1)^{{2+3}}\;\operatorname {det}{\begin{pmatrix}\!-3&\,2\\\,3&\!-4\end{pmatrix}}=-((-3)(-4)-(3)(2))=-6.$

Again, the (3,2) entry of the adjugate is the (2,3) cofactor of A. Thus, the submatrix

${\begin{pmatrix}\!-3&\,\!2\\\,\!3&\!-4\end{pmatrix}}$${\begin{pmatrix}\!-3&\,\!2\\\,\!3&\!-4\end{pmatrix}}$

was obtained by deleting the second row and third column of the original matrix A.

It is easy to check the adjugate is the inverse times the determinant, −6.

Properties

The adjugate has the properties

$\mathrm {adj} (\mathbf {I} )=\mathbf {I} ,$${\mathrm {adj}}({\mathbf {I}})={\mathbf {I}},$

$\mathrm {adj} (\mathbf {AB} )=\mathrm {adj} (\mathbf {B} )\,\mathrm {adj} (\mathbf {A} ),$${\mathrm {adj}}({\mathbf {AB}})={\mathrm {adj}}({\mathbf {B}})\,{\mathrm {adj}}({\mathbf {A}}),$

$\mathrm {adj} (c\mathbf {A} )=c^{n-1}\mathrm {adj} (\mathbf {A} )~,$$\mathrm{adj}(c \mathbf{A}) = c^{n - 1}\mathrm{adj}(\mathbf{A}) ~,$

for n×n matrices A and B. The second line follows from equations adj(B)adj(A) = det(B)B⁻¹ det(A)A⁻¹ = det(AB)(AB)⁻¹.

Substituting in the second line B = A^{m − 1} and performing the recursion, one finds, for all integer m,

$\mathrm {adj} (\mathbf {A} ^{m})=\mathrm {adj} (\mathbf {A} )^{m}~.$$\mathrm{adj}(\mathbf{A}^{m}) = \mathrm{adj}(\mathbf{A})^{m}~.$

The adjugate preserves transposition,

$\mathrm {adj} (\mathbf {A} ^{\mathsf {T}})=\mathrm {adj} (\mathbf {A} )^{\mathsf {T}}~.$$\mathrm{adj}(\mathbf{A}^\mathsf{T}) = \mathrm{adj}(\mathbf{A})^\mathsf{T}~.$

Furthermore,

If A is a n×n matrix with n ≥ 2, then $\det {\big (}\mathrm {adj} (\mathbf {A} ){\big )}=\det(\mathbf {A} )^{n-1},$$\det {\big (}{\mathrm {adj}}({\mathbf {A}}){\big )}=\det({\mathbf {A}})^{{n-1}},$ and

If A is an invertible n×n matrix, then $\mathrm {adj} (\mathrm {adj} (\mathbf {A} ))=\det(\mathbf {A} )^{n-2}\mathbf {A} ~,$$\mathrm{adj}(\mathrm{adj}(\mathbf{A})) = \det(\mathbf{A})^{n - 2}\mathbf{A} ~,$

so that, if n = 2 and A is invertible, then det(adj(A)) = det(A) and adj(adj(A)) = A.

Taking the adjugate of an invertible matrix A k times yields

$\mathrm {adj} _{k}(\mathbf {A} )=\det(\mathbf {A} )^{\frac {(n-1)^{k}-(-1)^{k}}{n}}\mathbf {A} ^{(-1)^{k}}~,$$\mathrm{adj}_k(\mathbf{A})=\det(\mathbf{A})^{\frac{(n-1)^k-(-1)^k}n}\mathbf{A}^{(-1)^k}~,$

$\det {\big (}\mathrm {adj} _{k}(\mathbf {A} ){\big )}=\det(\mathbf {A} )^{(n-1)^{k}}~.$$\det\big(\mathrm{adj}_k(\mathbf{A})\big)=\det(\mathbf{A})^{(n-1)^k}~.$

Inverses

In consequence of Laplace's formula for the determinant of an n×n matrix A,

$\mathbf {A} \,\mathrm {adj} (\mathbf {A} )=\mathrm {adj} (\mathbf {A} )\,\mathbf {A} =\det(\mathbf {A} )\,\mathbf {I} _{n}\qquad (*)$${\mathbf {A}}\,{\mathrm {adj}}({\mathbf {A}})={\mathrm {adj}}({\mathbf {A}})\,{\mathbf {A}}=\det({\mathbf {A}})\,{\mathbf I}_{n}\qquad (*)$

where $\mathbf {I} _{n}$${\mathbf I}_{n}$ is the n×n identity matrix. Indeed, the (i,i) entry of the product A adj(A) is the scalar product of row i of A with row i of the cofactor matrix C, which is simply the Laplace formula for det(A) expanded by row i.

Moreover, for i ≠ j the (i,j) entry of the product is the scalar product of row i of A with row j of C, which is the Laplace formula for the determinant of a matrix whose i and j rows are equal, and therefore vanishes.

From this formula follows one of the central results in matrix algebra: A matrix A over a commutative ring R is invertible if and only if det(A) is invertible in R.

For, if A is an invertible matrix, then

$1=\det(\mathbf {I} _{n})=\det(\mathbf {A} \mathbf {A} ^{-1})=\det(\mathbf {A} )\det(\mathbf {A} ^{-1})~,$$1=\det({\mathbf I}_{n})=\det({\mathbf {A}}{\mathbf {A}}^{{-1}})=\det({\mathbf {A}})\det({\mathbf {A}}^{{-1}})~,$

and equation (*) above implies

$\mathbf {A} ^{-1}=\det(\mathbf {A} )^{-1}\,\mathrm {adj} (\mathbf {A} )~.$$\mathbf{A}^{-1} = \det(\mathbf{A})^{-1}\, \mathrm{adj}(\mathbf{A})~.$

Similarly, the resolvent of A is

$R(t;\mathbf {A} )\equiv {\frac {\mathbf {I} }{t\mathbf {\mathbf {I} } -\mathbf {A} }}={\frac {\mathrm {adj} (t\mathbf {I} -\mathbf {A} )}{p(t)}}~,$$R(t;{\mathbf {A}})\equiv {\frac {{\mathbf {I}}}{t{\mathbf {{\mathbf {I}}}}-{\mathbf {A}}}}={\frac {{\mathrm {adj}}(t{\mathbf {I}}-{\mathbf {A}})}{p(t)}}~,$

where p(t) is the characteristic polynomial of A.

See also: Cramer's rule and Cayley–Hamilton theorem

Characteristic polynomial

If

$p(t)~{\stackrel {\text{def}}{=}}~\det(t\mathbf {I} -\mathbf {A} )=\sum _{i=0}^{n}p_{i}t^{i}\in R[t],$$p(t)~{\stackrel {\text{def}}{=}}~\det(t\mathbf {I} -\mathbf {A} )=\sum _{i=0}^{n}p_{i}t^{i}\in R[t],$

is the characteristic polynomial of the matrix n-by-n matrix $\mathbf {A}$$\mathbf {A}$ with coefficients in the ring R, then

$\mathrm {adj} \,(s\mathbf {I} -\mathbf {A} )=\mathrm {\Delta } \!p(s,\mathbf {A} )~,$$\mathrm{adj}\,(s\mathbf{I}-\mathbf{A}) = \mathrm{\Delta}\! p(s,\mathbf{A}) ~,$

where

$\mathrm {\Delta } \!p(s,t)~{\stackrel {\text{def}}{=}}~{\frac {p(s)-p(t)}{s-t}}=\sum _{j=0}^{n-1}\sum _{i=0}^{n-j-1}p_{i+j+1}s^{i}t^{j}\in R[s,t]$$\mathrm {\Delta } \!p(s,t)~{\stackrel {\text{def}}{=}}~{\frac {p(s)-p(t)}{s-t}}=\sum _{j=0}^{n-1}\sum _{i=0}^{n-j-1}p_{i+j+1}s^{i}t^{j}\in R[s,t]$

is the first divided difference of p, a symmetric polynomial of degree n−1.

Jacobi's formula

Main article: Jacobi's formula

The adjugate also appears in Jacobi's formula for the derivative of the determinant,

${\frac {\mathrm {d} }{\mathrm {d} \alpha }}\det(A)=\operatorname {tr} \left(\operatorname {adj} (A){\frac {\mathrm {d} A}{\mathrm {d} \alpha }}\right).$${\frac {{\mathrm {d}}}{{\mathrm {d}}\alpha }}\det(A)=\operatorname {tr}\left(\operatorname {adj}(A){\frac {{\mathrm {d}}A}{{\mathrm {d}}\alpha }}\right).$

Cayley–Hamilton formula

Main article: Cayley–Hamilton theorem

The Cayley–Hamilton theorem allows the adjugate of A to be represented in terms of traces and powers of A:

$\mathrm {adj} (\mathbf {A} )=\sum _{s=0}^{n-1}\mathbf {A} ^{s}\sum _{k_{1},k_{2},\ldots ,k_{n-1}}\prod _{l=1}^{n-1}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\mathrm {tr} (\mathbf {A} ^{l})^{k_{l}},$${\mathrm {adj}}({\mathbf {A}})=\sum _{{s=0}}^{{n-1}}{\mathbf {A}}^{{s}}\sum _{{k_{1},k_{2},\ldots ,k_{{n-1}}}}\prod _{{l=1}}^{{n-1}}{\frac {(-1)^{{k_{l}+1}}}{l^{{k_{l}}}k_{{l}}!}}{\mathrm {tr}}({\mathbf {A}}^{l})^{{k_{l}}},$

where n is the dimension of A, and the sum is taken over s and all sequences of k_l ≥ 0 satisfying the linear Diophantine equation

$s+\sum _{l=1}^{n-1}lk_{l}=n-1.$$s+\sum _{l=1}^{n-1}lk_{l}=n-1.$

For the 2×2 case this gives

$\mathrm {adj} (\mathbf {A} )=\mathbf {I} _{2}\mathrm {tr} \mathbf {A} -\mathbf {A} .$${\mathrm {adj}}({\mathbf {A}})={\mathbf {I}}_{2}{\mathrm {tr}}{\mathbf {A}}-{\mathbf {A}}.$

For the 3×3 case this gives

$\mathrm {adj} (\mathbf {A} )={\frac {1}{2}}\left((\mathrm {tr} \mathbf {A} )^{2}-\mathrm {tr} \mathbf {A} ^{2}\right)\mathbf {I} _{3}-\mathbf {A} \mathrm {tr} \mathbf {A} +\mathbf {A} ^{2}.$${\mathrm {adj}}({\mathbf {A}})={\frac {1}{2}}\left(({\mathrm {tr}}{\mathbf {A}})^{{2}}-{\mathrm {tr}}{\mathbf {A}}^{{2}}\right){\mathbf {I}}_{3}-{\mathbf {A}}{\mathrm {tr}}{\mathbf {A}}+{\mathbf {A}}^{{2}}.$

For the 4×4 case this gives

$\mathrm {adj} (\mathbf {A} )={\frac {1}{6}}\left((\mathrm {tr} \mathbf {A} )^{3}-3\mathrm {tr} \mathbf {A} \mathrm {tr} \mathbf {A} ^{2}+2\mathrm {tr} \mathbf {A} ^{3}\right)\mathbf {I} _{4}-{\frac {1}{2}}\mathbf {A} \left((\mathrm {tr} \mathbf {A} )^{2}-\mathrm {tr} \mathbf {A} ^{2}\right)+\mathbf {A} ^{2}\mathrm {tr} \mathbf {A} -\mathbf {A} ^{3}.$${\mathrm {adj}}({\mathbf {A}})={\frac {1}{6}}\left(({\mathrm {tr}}{\mathbf {A}})^{{3}}-3{\mathrm {tr}}{\mathbf {A}}{\mathrm {tr}}{\mathbf {A}}^{{2}}+2{\mathrm {tr}}{\mathbf {A}}^{{3}}\right){\mathbf {I}}_{4}-{\frac {1}{2}}{\mathbf {A}}\left(({\mathrm {tr}}{\mathbf {A}})^{{2}}-{\mathrm {tr}}{\mathbf {A}}^{{2}}\right)+{\mathbf {A}}^{{2}}{\mathrm {tr}}{\mathbf {A}}-{\mathbf {A}}^{{3}}.$

The same results follow directly from the terminating step of the fast Faddeev–LeVerrier algorithm.

See also

• Cayley–Hamilton theorem
• Cramer's rule
• Trace diagram
• Jacobi's formula
• Faddeev–LeVerrier algorithm

References

• Roger A. Horn and Charles R. Johnson (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1

External links

• Matrix Reference Manual
• Online matrix calculator (determinant, track, inverse, adjoint, transpose) Compute Adjugate matrix up to order 8
• "adjugate of { { a, b, c }, { d, e, f }, { g, h, i } }". Wolfram Alpha.