Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Adjugate matrix
open-in-new
<h2 id="definition">Definition</h2> <p>The adjugate of A is the <a href="/facts/Transpose/8wmsagGS">transpose</a> of the <a href="/facts/Cofactor_matrix/CMudF5Eh">cofactor matrix</a> C of A, </p> adj ( A ) = C T . {\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}.} <p>In more detail, suppose <i>R</i> is a <a href="/facts/Unital_algebra/8T8ulWJb">unital</a>+ <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> and A is an <i>n</i> × <i>n</i> matrix with entries from <i>R</i>. The (<i>i</i>, <i>j</i>)-<i><a href="/facts/Minor_(linear_algebra)/CMudF5Eh">minor</a></i> of A, denoted M<i>ij</i>, is the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the (<i>n</i> − 1) × (<i>n</i> − 1) matrix that results from deleting row i and column j of A. The <a href="/facts/Cofactor_(linear_algebra)/CMudF5Eh">cofactor matrix</a> of A is the <i>n</i> × <i>n</i> matrix C whose (<i>i</i>, <i>j</i>) entry is the (<i>i</i>, <i>j</i>) <i><a href="/facts/Cofactor_(linear_algebra)/CMudF5Eh">cofactor</a></i> of A, which is the (<i>i</i>, <i>j</i>)-minor times a sign factor: </p> C = ( ( − 1 ) i + j M i j ) 1 ≤ i , j ≤ n . {\displaystyle \mathbf {C} =\left((-1)^{i+j}\mathbf {M} _{ij}\right)_{1\leq i,j\leq n}.} <p>The adjugate of A is the transpose of C, that is, the <i>n</i> × <i>n</i> matrix whose (<i>i</i>, <i>j</i>) entry is the (<i>j</i>,<i>i</i>) cofactor of A, </p> adj ( A ) = C T = ( ( − 1 ) i + j M j i ) 1 ≤ i , j ≤ n . {\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}=\left((-1)^{i+j}\mathbf {M} _{ji}\right)_{1\leq i,j\leq n}.} <h3>Important consequence</h3> <p>The adjugate is defined so that the product of A with its adjugate yields a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> whose diagonal entries are the determinant det(A). That is, </p> A adj ( A ) = adj ( A ) A = det ( A ) I , {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )=\operatorname {adj} (\mathbf {A} )\mathbf {A} =\det(\mathbf {A} )\mathbf {I} ,} <p>where I is the <i>n</i> × <i>n</i> <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. This is a consequence of the <a href="/facts/Laplace_expansion/B4EIjlsp">Laplace expansion</a> of the determinant. </p><p>The above formula implies one of the fundamental results in matrix algebra, that A is <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> det(A) is an <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">invertible element</a> of <i>R</i>. When this holds, the equation above yields </p> adj ( A ) = det ( A ) A − 1 , A − 1 = det ( A ) − 1 adj ( A ) . {\displaystyle {\begin{aligned}\operatorname {adj} (\mathbf {A} )&=\det(\mathbf {A} )\mathbf {A} ^{-1},\\\mathbf {A} ^{-1}&=\det(\mathbf {A} )^{-1}\operatorname {adj} (\mathbf {A} ).\end{aligned}}} <h2 id="examples">Examples</h2> <h3>1 × 1 generic matrix</h3> <p>Since the determinant of a 0 × 0 matrix is 1, the adjugate of any 1 × 1 matrix (<a href="/facts/Complex_number/2w7mqI7e">complex</a> scalar) is I = [ 1 ] {\displaystyle \mathbf {I} ={\begin{bmatrix}1\end{bmatrix}}} . Observe that A adj ( A ) = adj ( A ) A = ( det A ) I . {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )=\operatorname {adj} (\mathbf {A} )\mathbf {A} =(\det \mathbf {A} )\mathbf {I} .} </p> <h3>2 × 2 generic matrix</h3> <p>The adjugate of the 2 × 2 matrix </p> A = [ a b c d ] {\displaystyle \mathbf {A} ={\begin{bmatrix}a&b\\c&d\end{bmatrix}}} <p>is </p> adj ( A ) = [ d − b − c a ] . {\displaystyle \operatorname {adj} (\mathbf {A} )={\begin{bmatrix}d&-b\\-c&a\end{bmatrix}}.} <p>By direct computation, </p> A adj ( A ) = [ a d − b c 0 0 a d − b c ] = ( det A ) I . {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} )={\begin{bmatrix}ad-bc&0\\0&ad-bc\end{bmatrix}}=(\det \mathbf {A} )\mathbf {I} .} <p>In this case, it is also true that det(adj(A)) = det(A) and hence that adj(adj(A)) = A. </p> <h3>3 × 3 generic matrix</h3> <p>Consider a 3 × 3 matrix </p> A = [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] . {\displaystyle \mathbf {A} ={\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{bmatrix}}.} <p>Its cofactor matrix is </p> C = [ + | b 2 b 3 c 2 c 3 | − | b 1 b 3 c 1 c 3 | + | b 1 b 2 c 1 c 2 | − | a 2 a 3 c 2 c 3 | + | a 1 a 3 c 1 c 3 | − | a 1 a 2 c 1 c 2 | + | a 2 a 3 b 2 b 3 | − | a 1 a 3 b 1 b 3 | + | a 1 a 2 b 1 b 2 | ] , {\displaystyle \mathbf {C} ={\begin{bmatrix}+{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}\\\\-{\begin{vmatrix}a_{2}&a_{3}\\c_{2}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{3}\\c_{1}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{2}\\c_{1}&c_{2}\end{vmatrix}}\\\\+{\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\end{bmatrix}},} <p>where </p> | a b c d | = det [ a b c d ] . {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=\det \!{\begin{bmatrix}a&b\\c&d\end{bmatrix}}.} <p>Its adjugate is the transpose of its cofactor matrix, </p> adj ( A ) = C T = [ + | b 2 b 3 c 2 c 3 | − | a 2 a 3 c 2 c 3 | + | a 2 a 3 b 2 b 3 | − | b 1 b 3 c 1 c 3 | + | a 1 a 3 c 1 c 3 | − | a 1 a 3 b 1 b 3 | + | b 1 b 2 c 1 c 2 | − | a 1 a 2 c 1 c 2 | + | a 1 a 2 b 1 b 2 | ] . {\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}={\begin{bmatrix}+{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{2}&a_{3}\\c_{2}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}\\&&\\-{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{3}\\c_{1}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\end{vmatrix}}\\&&\\+{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{2}\\c_{1}&c_{2}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\end{bmatrix}}.} <h3>3 × 3 numeric matrix</h3> <p>As a specific example, we have </p> adj [ − 3 2 − 5 − 1 0 − 2 3 − 4 1 ] = [ − 8 18 − 4 − 5 12 − 1 4 − 6 2 ] . {\displaystyle \operatorname {adj} \!{\begin{bmatrix}-3&2&-5\\-1&0&-2\\3&-4&1\end{bmatrix}}={\begin{bmatrix}-8&18&-4\\-5&12&-1\\4&-6&2\end{bmatrix}}.} <p>It is easy to check the adjugate is the <a href="/facts/Inverse_matrix/fPqXk3V8">inverse</a> times the determinant, −6. </p><p>The −1 in the second row, third column of the adjugate was computed as follows. The (2,3) entry of the adjugate is the (3,2) cofactor of A. This cofactor is computed using the <a href="/facts/Submatrix/qa8FI4ko">submatrix</a> obtained by deleting the third row and second column of the original matrix A, </p> [ − 3 − 5 − 1 − 2 ] . {\displaystyle {\begin{bmatrix}-3&-5\\-1&-2\end{bmatrix}}.} <p>The (3,2) cofactor is a sign times the determinant of this submatrix: </p> ( − 1 ) 3 + 2 det [ − 3 − 5 − 1 − 2 ] = − ( − 3 ⋅ − 2 − − 5 ⋅ − 1 ) = − 1 , {\displaystyle (-1)^{3+2}\operatorname {det} \!{\begin{bmatrix}-3&-5\\-1&-2\end{bmatrix}}=-(-3\cdot -2--5\cdot -1)=-1,} <p>and this is the (2,3) entry of the adjugate. </p> <h2 id="properties">Properties</h2> <p>For any <i>n</i> × <i>n</i> matrix A, elementary computations show that adjugates have the following properties: </p> <ul><li> adj ( I ) = I {\displaystyle \operatorname {adj} (\mathbf {I} )=\mathbf {I} } , where I {\displaystyle \mathbf {I} } is the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>.</li> <li> adj ( 0 ) = 0 {\displaystyle \operatorname {adj} (\mathbf {0} )=\mathbf {0} } , where 0 {\displaystyle \mathbf {0} } is the <a href="/facts/Zero_matrix/uqbGFFww">zero matrix</a>, except that if n = 1 {\displaystyle n=1} then adj ( 0 ) = I {\displaystyle \operatorname {adj} (\mathbf {0} )=\mathbf {I} } .</li> <li> adj ( c A ) = c n − 1 adj ( A ) {\displaystyle \operatorname {adj} (c\mathbf {A} )=c^{n-1}\operatorname {adj} (\mathbf {A} )} for any scalar c.</li> <li> adj ( A T ) = adj ( A ) T {\displaystyle \operatorname {adj} (\mathbf {A} ^{\mathsf {T}})=\operatorname {adj} (\mathbf {A} )^{\mathsf {T}}} .</li> <li> det ( adj ( A ) ) = ( det A ) n − 1 {\displaystyle \det(\operatorname {adj} (\mathbf {A} ))=(\det \mathbf {A} )^{n-1}} .</li> <li>If A is invertible, then adj ( A ) = ( det A ) A − 1 {\displaystyle \operatorname {adj} (\mathbf {A} )=(\det \mathbf {A} )\mathbf {A} ^{-1}} . It follows that: <ul><li>adj(A) is invertible with inverse (det A)−1A.</li> <li>adj(A−1) = adj(A)−1.</li></ul></li> <li>adj(A) is entrywise <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> in A. In particular, over the <a href="/facts/Real_number/R02gw5Pb">real</a> or complex numbers, the adjugate is a <a href="/facts/Smooth_function/mffL4ch2">smooth function</a> of the entries of A.</li></ul> <p>Over the complex numbers, </p> <ul><li> adj ( A ¯ ) = adj ( A ) ¯ {\displaystyle \operatorname {adj} ({\overline {\mathbf {A} }})={\overline {\operatorname {adj} (\mathbf {A} )}}} , where the bar denotes <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a>.</li> <li> adj ( A ∗ ) = adj ( A ) ∗ {\displaystyle \operatorname {adj} (\mathbf {A} ^{*})=\operatorname {adj} (\mathbf {A} )^{*}} , where the asterisk denotes <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a>.</li></ul> <p>Suppose that B is another <i>n</i> × <i>n</i> matrix. Then </p> adj ( A B ) = adj ( B ) adj ( A ) . {\displaystyle \operatorname {adj} (\mathbf {AB} )=\operatorname {adj} (\mathbf {B} )\operatorname {adj} (\mathbf {A} ).} <p>This can be <a href="/facts/Mathematical_proof/7ECDrU80">proved</a> in three ways. One way, valid for any commutative ring, is a direct computation using the <a href="/facts/Cauchy%E2%80%93Binet_formula/jbRCH0tj">Cauchy–Binet formula</a>. The second way, valid for the real or complex numbers, is to first observe that for invertible matrices A and B, </p> adj ( B ) adj ( A ) = ( det B ) B − 1 ( det A ) A − 1 = ( det A B ) ( A B ) − 1 = adj ( A B ) . {\displaystyle \operatorname {adj} (\mathbf {B} )\operatorname {adj} (\mathbf {A} )=(\det \mathbf {B} )\mathbf {B} ^{-1}(\det \mathbf {A} )\mathbf {A} ^{-1}=(\det \mathbf {AB} )(\mathbf {AB} )^{-1}=\operatorname {adj} (\mathbf {AB} ).} <p>Because every non-invertible matrix is the limit of invertible matrices, <a href="/facts/Continuous_function/sKbl02pB">continuity</a> of the adjugate then implies that the formula remains true when one of A or B is not invertible. </p><p>A <a href="/facts/Corollary/ntQ8zR9N">corollary</a> of the previous formula is that, for any non-negative <a href="/facts/Integer/PUwwrawx">integer</a> k, </p> adj ( A k ) = adj ( A ) k . {\displaystyle \operatorname {adj} (\mathbf {A} ^{k})=\operatorname {adj} (\mathbf {A} )^{k}.} <p>If A is invertible, then the above formula also holds for negative k. </p><p>From the identity </p> ( A + B ) adj ( A + B ) B = det ( A + B ) B = B adj ( A + B ) ( A + B ) , {\displaystyle (\mathbf {A} +\mathbf {B} )\operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {B} =\det(\mathbf {A} +\mathbf {B} )\mathbf {B} =\mathbf {B} \operatorname {adj} (\mathbf {A} +\mathbf {B} )(\mathbf {A} +\mathbf {B} ),} <p>we deduce </p> A adj ( A + B ) B = B adj ( A + B ) A . {\displaystyle \mathbf {A} \operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {B} =\mathbf {B} \operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {A} .} <p>Suppose that A <a href="/facts/Commuting_matrices/R1wGesmf">commutes</a> with B. Multiplying the identity AB = BA on the left and right by adj(A) proves that </p> det ( A ) adj ( A ) B = det ( A ) B adj ( A ) . {\displaystyle \det(\mathbf {A} )\operatorname {adj} (\mathbf {A} )\mathbf {B} =\det(\mathbf {A} )\mathbf {B} \operatorname {adj} (\mathbf {A} ).} <p>If A is invertible, this implies that adj(A) also commutes with B. Over the real or complex numbers, continuity implies that adj(A) commutes with B even when A is not invertible. </p><p>Finally, there is a more general proof than the second proof, which only requires that an <i>n</i> × <i>n</i> matrix has entries over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> with at least 2<i>n</i> + 1 elements (e.g. a 5 × 5 matrix over the integers <a href="/facts/Modular_arithmetic/0wtRa5dU">modulo</a> 11). det(A+<i>t</i>I) is a polynomial in <i>t</i> with <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> at most <i>n</i>, so it has at most <i>n</i> <a href="/facts/Root_of_a_polynomial/mlK3dhoT">roots</a>. Note that the <i>ij</i>th entry of adj((A+<i>t</i>I)(B)) is a polynomial of at most order <i>n</i>, and likewise for adj(A+<i>t</i>I)adj(B). These two polynomials at the <i>ij</i>th entry agree on at least <i>n</i> + 1 points, as we have at least <i>n</i> + 1 elements of the field where A+<i>t</i>I is invertible, and we have proven the identity for invertible matrices. Polynomials of degree <i>n</i> which agree on <i>n</i> + 1 points must be identical (subtract them from each other and you have <i>n</i> + 1 roots for a polynomial of degree at most <i>n</i> – a contradiction unless their difference is identically zero). As the two polynomials are identical, they take the same value for every value of <i>t</i>. Thus, they take the same value when <i>t</i> = 0. </p><p>Using the above properties and other elementary computations, it is straightforward to show that if A has one of the following properties, then adjA does as well: </p> <ul><li><a href="/facts/Upper_triangular_matrix/qjXSFMM3">upper triangular</a>,</li> <li><a href="/facts/Lower_triangular_matrix/qjXSFMM3">lower triangular</a>,</li> <li><a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal</a>,</li> <li><a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal</a>,</li> <li><a href="/facts/Unitary_matrix/8fsxv3KD">unitary</a>,</li> <li><a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a>,</li> <li><a href="/facts/Hermitian_matrix/qArPblYo">Hermitian</a>,</li> <li><a href="/facts/Normal_matrix/WVKxrEK9">normal</a>.</li></ul> <p>If A is <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">skew-symmetric</a>, then adj(A) is skew-symmetric for even <i>n</i> and symmetric for odd <i>n</i>. Similarly, if A is <a href="/facts/Skew-Hermitian_matrix/wxjjnvCC">skew-Hermitian</a>, then adj(A) is skew-Hermitian for even <i>n</i> and Hermitian for odd <i>n</i>. </p><p>If A is invertible, then, as noted above, there is a formula for adj(A) in terms of the determinant and inverse of A. When A is not invertible, the adjugate satisfies different but closely related formulas. </p> <ul><li>If rk(A) ≤ <i>n</i> − 2, then adj(A) = 0.</li> <li>If rk(A) = <i>n</i> − 1, then rk(adj(A)) = 1. (Some minor is non-zero, so adj(A) is non-zero and hence has <a href="/facts/Rank_(linear_algebra)/Yp9b5LK3">rank</a> at least one; the identity adj(A)A = 0 implies that the <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> of the <a href="/facts/Nullspace/adhGUYa0">nullspace</a> of adj(A) is at least <i>n</i> − 1, so its rank is at most one.) It follows that adj(A) = <i>α</i>xyT, where <i>α</i> is a scalar and x and y are vectors such that Ax = 0 and AT y = 0.</li></ul> <h3>Column substitution and Cramer's rule</h3> <p class="note">See also: <a href="/facts/Cramer%27s_rule/aUt4Cena">Cramer's rule</a></p> <p>Partition A into <a href="/facts/Column_vector/pPuRr9Xk">column vectors</a>: </p> A = [ a 1 ⋯ a n ] . {\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {a} _{1}&\cdots &\mathbf {a} _{n}\end{bmatrix}}.} <p>Let b be a column vector of size <i>n</i>. Fix 1 ≤ <i>i</i> ≤ <i>n</i> and consider the matrix formed by replacing column <i>i</i> of A by b: </p> ( A ← i b ) = def [ a 1 ⋯ a i − 1 b a i + 1 ⋯ a n ] . {\displaystyle (\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )\ {\stackrel {\text{def}}{=}}\ {\begin{bmatrix}\mathbf {a} _{1}&\cdots &\mathbf {a} _{i-1}&\mathbf {b} &\mathbf {a} _{i+1}&\cdots &\mathbf {a} _{n}\end{bmatrix}}.} <p>Laplace expand the determinant of this matrix along column i. The result is entry i of the product adj(A)b. Collecting these determinants for the different possible i yields an equality of column vectors </p> ( det ( A ← i b ) ) i = 1 n = adj ( A ) b . {\displaystyle \left(\det(\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )\right)_{i=1}^{n}=\operatorname {adj} (\mathbf {A} )\mathbf {b} .} <p>This formula has the following concrete consequence. Consider the <a href="/facts/Linear_system_of_equations/fk9CFdgp">linear system of equations</a> </p> A x = b . {\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} .} <p>Assume that A is <a href="/facts/Singular_matrix/fPqXk3V8">non-singular</a>. Multiplying this system on the left by adj(A) and dividing by the determinant yields </p> x = adj ( A ) b det A . {\displaystyle \mathbf {x} ={\frac {\operatorname {adj} (\mathbf {A} )\mathbf {b} }{\det \mathbf {A} }}.} <p>Applying the previous formula to this situation yields Cramer's rule, </p> x i = det ( A ← i b ) det A , {\displaystyle x_{i}={\frac {\det(\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )}{\det \mathbf {A} }},} <p>where <i>x</i><i>i</i> is the ith entry of x. </p> <h3>Characteristic polynomial</h3> <p>Let the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of A be </p> p ( s ) = det ( s I − A ) = ∑ i = 0 n p i s i ∈ R [ s ] . {\displaystyle p(s)=\det(s\mathbf {I} -\mathbf {A} )=\sum _{i=0}^{n}p_{i}s^{i}\in R[s].} <p>The first <a href="/facts/Divided_difference/HUIoAusz">divided difference</a> of <i>p</i> is a <a href="/facts/Symmetric_polynomial/GNhguVcn">symmetric polynomial</a> of degree <i>n</i> − 1, </p> Δ p ( s , t ) = p ( s ) − p ( t ) s − t = ∑ 0 ≤ j + k < n p j + k + 1 s j t k ∈ R [ s , t ] . {\displaystyle \Delta p(s,t)={\frac {p(s)-p(t)}{s-t}}=\sum _{0\leq j+k<n}p_{j+k+1}s^{j}t^{k}\in R[s,t].} <p>Multiply <i>s</i>I − A by its adjugate. Since <i>p</i>(A) = 0 by the <a href="/facts/Cayley%E2%80%93Hamilton_theorem/pXdJXWkR">Cayley–Hamilton theorem</a>, some elementary manipulations reveal </p> adj ( s I − A ) = Δ p ( s I , A ) . {\displaystyle \operatorname {adj} (s\mathbf {I} -\mathbf {A} )=\Delta p(s\mathbf {I} ,\mathbf {A} ).} <p>In particular, the <a href="/facts/Resolvent_formalism/IKaKtcef">resolvent</a> of A is defined to be </p> R ( z ; A ) = ( z I − A ) − 1 , {\displaystyle R(z;\mathbf {A} )=(z\mathbf {I} -\mathbf {A} )^{-1},} <p>and by the above formula, this is equal to </p> R ( z ; A ) = Δ p ( z I , A ) p ( z ) . {\displaystyle R(z;\mathbf {A} )={\frac {\Delta p(z\mathbf {I} ,\mathbf {A} )}{p(z)}}.} <h3>Jacobi's formula</h3> <p class="note">Main article: <a href="/facts/Jacobi%27s_formula/RhxPLByr">Jacobi's formula</a></p> <p>The adjugate also appears in <a href="/facts/Jacobi%27s_formula/RhxPLByr">Jacobi's formula</a> for the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of the determinant. If A(<i>t</i>) is <a href="/facts/Continuously_differentiable/jQqTgLk1">continuously differentiable</a>, then </p> d ( det A ) d t ( t ) = tr ( adj ( A ( t ) ) A ′ ( t ) ) . {\displaystyle {\frac {d(\det \mathbf {A} )}{dt}}(t)=\operatorname {tr} \left(\operatorname {adj} (\mathbf {A} (t))\mathbf {A} '(t)\right).} <p>It follows that the <a href="/facts/Total_derivative/cPwsm9Wf">total derivative</a> of the determinant is the transpose of the adjugate: </p> d ( det A ) A 0 = adj ( A 0 ) T . {\displaystyle d(\det \mathbf {A} )_{\mathbf {A} _{0}}=\operatorname {adj} (\mathbf {A} _{0})^{\mathsf {T}}.} <h3>Cayley–Hamilton formula</h3> <p class="note">Main article: <a href="/facts/Cayley%E2%80%93Hamilton_theorem/pXdJXWkR">Cayley–Hamilton theorem</a></p> <p>Let <i>p</i>A(<i>t</i>) be the characteristic polynomial of A. The <a href="/facts/Cayley%E2%80%93Hamilton_theorem/pXdJXWkR">Cayley–Hamilton theorem</a> states that </p> p A ( A ) = 0 . {\displaystyle p_{\mathbf {A} }(\mathbf {A} )=\mathbf {0} .} <p>Separating the constant term and multiplying the equation by adj(A) gives an expression for the adjugate that depends only on A and the coefficients of <i>p</i>A(<i>t</i>). These coefficients can be explicitly represented in terms of <a href="/facts/Trace_(linear_algebra)/dchFFvHt">traces</a> of powers of A using complete exponential <a href="/facts/Bell_polynomials/qQ9rxxbA">Bell polynomials</a>. The resulting formula is </p> adj ( A ) = ∑ s = 0 n − 1 A s ∑ k 1 , k 2 , … , k n − 1 ∏ ℓ = 1 n − 1 ( − 1 ) k ℓ + 1 ℓ k ℓ k ℓ ! tr ( A ℓ ) k ℓ , {\displaystyle \operatorname {adj} (\mathbf {A} )=\sum _{s=0}^{n-1}\mathbf {A} ^{s}\sum _{k_{1},k_{2},\ldots ,k_{n-1}}\prod _{\ell =1}^{n-1}{\frac {(-1)^{k_{\ell }+1}}{\ell ^{k_{\ell }}k_{\ell }!}}\operatorname {tr} (\mathbf {A} ^{\ell })^{k_{\ell }},} <p>where n is the dimension of A, and the sum is taken over s and all sequences of <i>kl</i> ≥ 0 satisfying the linear <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equation</a> </p> s + ∑ ℓ = 1 n − 1 ℓ k ℓ = n − 1. {\displaystyle s+\sum _{\ell =1}^{n-1}\ell k_{\ell }=n-1.} <p>For the 2 × 2 case, this gives </p> adj ( A ) = I 2 ( tr A ) − A . {\displaystyle \operatorname {adj} (\mathbf {A} )=\mathbf {I} _{2}(\operatorname {tr} \mathbf {A} )-\mathbf {A} .} <p>For the 3 × 3 case, this gives </p> adj ( A ) = 1 2 I 3 ( ( tr A ) 2 − tr A 2 ) − A ( tr A ) + A 2 . {\displaystyle \operatorname {adj} (\mathbf {A} )={\frac {1}{2}}\mathbf {I} _{3}\!\left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)-\mathbf {A} (\operatorname {tr} \mathbf {A} )+\mathbf {A} ^{2}.} <p>For the 4 × 4 case, this gives </p> adj ( A ) = 1 6 I 4 ( ( tr A ) 3 − 3 tr A tr A 2 + 2 tr A 3 ) − 1 2 A ( ( tr A ) 2 − tr A 2 ) + A 2 ( tr A ) − A 3 . {\displaystyle \operatorname {adj} (\mathbf {A} )={\frac {1}{6}}\mathbf {I} _{4}\!\left((\operatorname {tr} \mathbf {A} )^{3}-3\operatorname {tr} \mathbf {A} \operatorname {tr} \mathbf {A} ^{2}+2\operatorname {tr} \mathbf {A} ^{3}\right)-{\frac {1}{2}}\mathbf {A} \!\left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)+\mathbf {A} ^{2}(\operatorname {tr} \mathbf {A} )-\mathbf {A} ^{3}.} <p>The same formula follows directly from the terminating step of the <a href="/facts/Faddeev%E2%80%93LeVerrier_algorithm/uuKcU5tn">Faddeev–LeVerrier algorithm</a>, which efficiently determines the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of A. </p><p>In general, adjugate matrix of arbitrary dimension N matrix can be computed by Einstein's convention. </p> ( adj ( A ) ) i N j N = 1 ( N − 1 ) ! ϵ i 1 i 2 … i N ϵ j 1 j 2 … j N A j 1 i 1 A j 2 i 2 … A j N − 1 i N − 1 {\displaystyle (\operatorname {adj} (\mathbf {A} ))_{i_{N}}^{j_{N}}={\frac {1}{(N-1)!}}\epsilon _{i_{1}i_{2}\ldots i_{N}}\epsilon ^{j_{1}j_{2}\ldots j_{N}}A_{j_{1}}^{i_{1}}A_{j_{2}}^{i_{2}}\ldots A_{j_{N-1}}^{i_{N-1}}} <h2 id="relation-to-exterior-algebras">Relation to exterior algebras</h2> <p>The adjugate can be viewed in abstract terms using <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebras</a>. Let <i>V</i> be an <i>n</i>-dimensional <a href="/facts/Vector_space/M1pxMLx2">vector space</a>. The <a href="/facts/Exterior_product/rdN4TvpR">exterior product</a> defines a bilinear pairing V × ∧ n − 1 V → ∧ n V . {\displaystyle V\times \wedge ^{n-1}V\to \wedge ^{n}V.} Abstractly, ∧ n V {\displaystyle \wedge ^{n}V} is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to R, and under any such isomorphism the exterior product is a <a href="/facts/Perfect_pairing/gyvzn9ym">perfect pairing</a>. That is, it yields an isomorphism ϕ : V → ≅ Hom ( ∧ n − 1 V , ∧ n V ) . {\displaystyle \phi \colon V\ {\xrightarrow {\cong }}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V).} This isomorphism sends each v ∈ <i>V</i> to the map ϕ v {\displaystyle \phi _{\mathbf {v} }} defined by ϕ v ( α ) = v ∧ α . {\displaystyle \phi _{\mathbf {v} }(\alpha )=\mathbf {v} \wedge \alpha .} Suppose that <i>T</i> : <i>V</i> → <i>V</i> is a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a>. <a href="/facts/Pullback/AcMfwboF">Pullback</a> by the (<i>n</i> − 1)th exterior power of <i>T</i> induces a morphism of Hom spaces. The adjugate of <i>T</i> is the composite V → ϕ Hom ( ∧ n − 1 V , ∧ n V ) → ( ∧ n − 1 T ) ∗ Hom ( ∧ n − 1 V , ∧ n V ) → ϕ − 1 V . {\displaystyle V\ {\xrightarrow {\phi }}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V)\ {\xrightarrow {(\wedge ^{n-1}T)^{*}}}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V)\ {\xrightarrow {\phi ^{-1}}}\ V.} </p><p>If <i>V</i> = R<i>n</i> is endowed with its <a href="/facts/Canonical_basis/I3dJq0Q3">canonical basis</a> e1, ..., e<i>n</i>, and if the matrix of <i>T</i> in this <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> is A, then the adjugate of <i>T</i> is the adjugate of A. To see why, give ∧ n − 1 R n {\displaystyle \wedge ^{n-1}\mathbf {R} ^{n}} the basis { e 1 ∧ ⋯ ∧ e ^ k ∧ ⋯ ∧ e n } k = 1 n . {\displaystyle \{\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n}\}_{k=1}^{n}.} Fix a basis vector e<i>i</i> of R<i>n</i>. The image of e<i>i</i> under ϕ {\displaystyle \phi } is determined by where it sends basis vectors: ϕ e i ( e 1 ∧ ⋯ ∧ e ^ k ∧ ⋯ ∧ e n ) = { ( − 1 ) i − 1 e 1 ∧ ⋯ ∧ e n , if k = i , 0 otherwise. {\displaystyle \phi _{\mathbf {e} _{i}}(\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n})={\begin{cases}(-1)^{i-1}\mathbf {e} _{1}\wedge \dots \wedge \mathbf {e} _{n},&{\text{if}}\ k=i,\\0&{\text{otherwise.}}\end{cases}}} On basis vectors, the (<i>n</i> − 1)st exterior power of <i>T</i> is e 1 ∧ ⋯ ∧ e ^ j ∧ ⋯ ∧ e n ↦ ∑ k = 1 n ( det A j k ) e 1 ∧ ⋯ ∧ e ^ k ∧ ⋯ ∧ e n . {\displaystyle \mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{j}\wedge \dots \wedge \mathbf {e} _{n}\mapsto \sum _{k=1}^{n}(\det A_{jk})\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n}.} Each of these terms maps to zero under ϕ e i {\displaystyle \phi _{\mathbf {e} _{i}}} except the <i>k</i> = <i>i</i> term. Therefore, the pullback of ϕ e i {\displaystyle \phi _{\mathbf {e} _{i}}} is the linear transformation for which e 1 ∧ ⋯ ∧ e ^ j ∧ ⋯ ∧ e n ↦ ( − 1 ) i − 1 ( det A j i ) e 1 ∧ ⋯ ∧ e n . {\displaystyle \mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{j}\wedge \dots \wedge \mathbf {e} _{n}\mapsto (-1)^{i-1}(\det A_{ji})\mathbf {e} _{1}\wedge \dots \wedge \mathbf {e} _{n}.} That is, it equals ∑ j = 1 n ( − 1 ) i + j ( det A j i ) ϕ e j . {\displaystyle \sum _{j=1}^{n}(-1)^{i+j}(\det A_{ji})\phi _{\mathbf {e} _{j}}.} Applying the inverse of ϕ {\displaystyle \phi } shows that the adjugate of <i>T</i> is the linear transformation for which e i ↦ ∑ j = 1 n ( − 1 ) i + j ( det A j i ) e j . {\displaystyle \mathbf {e} _{i}\mapsto \sum _{j=1}^{n}(-1)^{i+j}(\det A_{ji})\mathbf {e} _{j}.} Consequently, its matrix representation is the adjugate of A. </p><p>If <i>V</i> is endowed with an <a href="/facts/Inner_product/HoyjElEy">inner product</a> and a volume form, then the map <i>φ</i> can be decomposed further. In this case, <i>φ</i> can be understood as the composite of the <a href="/facts/Hodge_star_operator/AH6tmHhU">Hodge star operator</a> and dualization. Specifically, if ω is the volume form, then it, together with the inner product, determines an isomorphism ω ∨ : ∧ n V → R . {\displaystyle \omega ^{\vee }\colon \wedge ^{n}V\to \mathbf {R} .} This induces an isomorphism Hom ( ∧ n − 1 R n , ∧ n R n ) ≅ ∧ n − 1 ( R n ) ∨ . {\displaystyle \operatorname {Hom} (\wedge ^{n-1}\mathbf {R} ^{n},\wedge ^{n}\mathbf {R} ^{n})\cong \wedge ^{n-1}(\mathbf {R} ^{n})^{\vee }.} A vector v in R<i>n</i> corresponds to the linear functional ( α ↦ ω ∨ ( v ∧ α ) ) ∈ ∧ n − 1 ( R n ) ∨ . {\displaystyle (\alpha \mapsto \omega ^{\vee }(\mathbf {v} \wedge \alpha ))\in \wedge ^{n-1}(\mathbf {R} ^{n})^{\vee }.} By the definition of the Hodge star operator, this linear functional is dual to *v. That is, ω∨∘ φ equals v ↦ *v∨. </p> <h2 id="higher-adjugates">Higher adjugates</h2> <p>Let A be an <i>n</i> × <i>n</i> matrix, and fix <i>r</i> ≥ 0. The <i>r</i>th higher adjugate of A is an ( n r ) × ( n r ) {\textstyle {\binom {n}{r}}\!\times \!{\binom {n}{r}}} matrix, denoted adj<i>r</i> A, whose entries are indexed by size <i>r</i> <a href="/facts/Subset/SJGERmbA">subsets</a> <i>I</i> and <i>J</i> of {1, ..., <i>m</i>} . Let <i>I</i>c and <i>J</i>c denote the <a href="/facts/Complement_(set_theory)/yWUePIYY">complements</a> of <i>I</i> and <i>J</i>, respectively. Also let A I c , J c {\displaystyle \mathbf {A} _{I^{c},J^{c}}} denote the submatrix of A containing those rows and columns whose indices are in <i>I</i>c and <i>J</i>c, respectively. Then the (<i>I</i>, <i>J</i>) entry of adj<i>r</i> A is </p> ( − 1 ) σ ( I ) + σ ( J ) det A J c , I c , {\displaystyle (-1)^{\sigma (I)+\sigma (J)}\det \mathbf {A} _{J^{c},I^{c}},} <p>where σ(<i>I</i>) and σ(<i>J</i>) are the sum of the elements of <i>I</i> and <i>J</i>, respectively. </p><p>Basic properties of higher adjugates include : </p> <ul><li>adj0(A) = det A.</li> <li>adj1(A) = adj A.</li> <li>adj<i>n</i>(A) = 1.</li> <li>adj<i>r</i>(BA) = adj<i>r</i>(A) adj<i>r</i>(B).</li> <li> adj r ( A ) C r ( A ) = C r ( A ) adj r ( A ) = ( det A ) I ( n r ) {\displaystyle \operatorname {adj} _{r}(\mathbf {A} )C_{r}(\mathbf {A} )=C_{r}(\mathbf {A} )\operatorname {adj} _{r}(\mathbf {A} )=(\det \mathbf {A} )I_{\binom {n}{r}}} , where <i>C</i><i>r</i>(A) denotes the <i>r</i>th <a href="/facts/Compound_matrix/06IhNVUN">compound matrix</a>.</li></ul> <p>Higher adjugates may be defined in abstract algebraic terms in a similar fashion to the usual adjugate, substituting ∧ r V {\displaystyle \wedge ^{r}V} and ∧ n − r V {\displaystyle \wedge ^{n-r}V} for V {\displaystyle V} and ∧ n − 1 V {\displaystyle \wedge ^{n-1}V} , respectively. </p> <h2 id="iterated-adjugates">Iterated adjugates</h2> <p><a href="/facts/Iterated_function/Vzsx64An">Iteratively</a> taking the adjugate of an invertible matrix A k times yields </p> adj ⋯ adj ⏞ k ( A ) = det ( A ) ( n − 1 ) k − ( − 1 ) k n A ( − 1 ) k , {\displaystyle \overbrace {\operatorname {adj} \dotsm \operatorname {adj} } ^{k}(\mathbf {A} )=\det(\mathbf {A} )^{\frac {(n-1)^{k}-(-1)^{k}}{n}}\mathbf {A} ^{(-1)^{k}},} det ( adj ⋯ adj ⏞ k ( A ) ) = det ( A ) ( n − 1 ) k . {\displaystyle \det(\overbrace {\operatorname {adj} \dotsm \operatorname {adj} } ^{k}(\mathbf {A} ))=\det(\mathbf {A} )^{(n-1)^{k}}.} <p>For example, </p> adj ( adj ( A ) ) = det ( A ) n − 2 A . {\displaystyle \operatorname {adj} (\operatorname {adj} (\mathbf {A} ))=\det(\mathbf {A} )^{n-2}\mathbf {A} .} det ( adj ( adj ( A ) ) ) = det ( A ) ( n − 1 ) 2 . {\displaystyle \det(\operatorname {adj} (\operatorname {adj} (\mathbf {A} )))=\det(\mathbf {A} )^{(n-1)^{2}}.} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Cayley%E2%80%93Hamilton_theorem/pXdJXWkR">Cayley–Hamilton theorem</a></li> <li><a href="/facts/Cramer%27s_rule/aUt4Cena">Cramer's rule</a></li> <li><a href="/facts/Trace_diagram/y32UApwR">Trace diagram</a></li> <li><a href="/facts/Jacobi%27s_formula/RhxPLByr">Jacobi's formula</a></li> <li><a href="/facts/Faddeev%E2%80%93LeVerrier_algorithm/uuKcU5tn">Faddeev–LeVerrier algorithm</a></li> <li><a href="/facts/Compound_matrix/06IhNVUN">Compound matrix</a></li></ul> <h2 id="bibliography">Bibliography</h2> <ul><li>Roger A. Horn and Charles R. Johnson (2013), <i>Matrix Analysis</i>, Second Edition. Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-54823-6</li> <li>Roger A. Horn and Charles R. Johnson (1991), <i>Topics in Matrix Analysis</i>. Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-46713-1</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/property.html#adjoint">Matrix Reference Manual</a></li> <li><a href="http://www.elektro-energetika.cz/calculations/matreg.php?language=english">Online matrix calculator (determinant, track, inverse, adjoint, transpose)</a> Compute Adjugate matrix up to order 8</li> <li><a href="http://www.wolframalpha.com/input/?i=adjugate+of+{+{+a%2C+b%2C+c+}%2C+{+d%2C+e%2C+f+}%2C+{+g%2C+h%2C+i+}+}">"Adjugate of { { a, b, c }, { d, e, f }, { g, h, i } }"</a>. <i><a href="/facts/Wolfram_Alpha/CIBhWgXA">Wolfram Alpha</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Gantmacher, F. R. (1960). The Theory of Matrices. Vol. 1. New York: Chelsea. pp. 76–89. ISBN 0-8218-1376-5. <a href="0-8218-1376-5" target="_blank">0-8218-1376-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Strang, Gilbert (1988). "Section 4.4: Applications of determinants". Linear Algebra and its Applications (3rd ed.). Harcourt Brace Jovanovich. pp. 231–232. ISBN 0-15-551005-3. <a href="0-15-551005-3" target="_blank">0-15-551005-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Claeyssen, J.C.R. (1990). "On predicting the response of non-conservative linear vibrating systems by using dynamical matrix solutions". Journal of Sound and Vibration. 140 (1): 73–84. Bibcode:1990JSV...140...73C. doi:10.1016/0022-460X(90)90907-H. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Chen, W.; Chen, W.; Chen, Y.J. (2004). "A characteristic matrix approach for analyzing resonant ring lattice devices". IEEE Photonics Technology Letters. 16 (2): 458–460. Bibcode:2004IPTL...16..458C. doi:10.1109/LPT.2003.823104. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Householder, Alston S. (2006). The Theory of Matrices in Numerical Analysis. Dover Books on Mathematics. pp. 166–168. ISBN 0-486-44972-6. <a href="0-486-44972-6" target="_blank">0-486-44972-6</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>