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 . The Adjoint of 3×3 Matrix block computes the adjoint matrix for the input matrix. Calculating the inverse of a 3×3 matrix by hand is a tedious job, but worth reviewing. You can also find the This is sometimes referred to as the adjoint matrix.
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In linear algebrathe adjugateclassical adjointor 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 operatorwhich is its conjugate transpose. The adjugate of A is the transpose of the cofactor matrix C of A.
Inverse of a Matrix using Minors, Cofactors and Adjugate
The adjugate is defined as it is so that the product of A with its adjugate yields a diagonal matrix whose diagonal entries are the determinant det A. This is a consequence of the Laplace expansion of the determinant.
The above formula implies one of the fundamental results in matrix algebra, that A is invertible if and only if det A is an invertible element of Xdjoint. When this holds, the equation above yields. The 2,3 entry of the adjugate is the 3,2 cofactor of A.
This cofactor is computed using the afjoint obtained by deleting the third row and second column of the original matrix A.
This can be proved in two ways. One way, valid for any commutative ring, is a direct computation using the Cauchy—Binet formula. The other x33, valid for the real or complex numbers, is to first observe that for invertible matrices A and B.
Because every non-invertible matrix is the limit of invertible matrices, continuity of the adjugate then implies that the formula remains true when matrid of A or B is not invertible. A corollary of the previous formula is that, for any non-negative integer k. If A is invertible, then the above formula also holds for negative k. Suppose that A commutes with B.
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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. Using the above properties and other elementary computations, it is straightforward to show that if A has one of the following properties, then adj A does as well:.
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.
Let b adjiint a column vector of size n. 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. Assume that A is non-singular.
Multiplying this system on the left by adj A and dividing by the determinant yields. Let the characteristic polynomial of A be. In particular, the resolvent of A is defined to be. The adjugate also appears in Jacobi’s formula for the derivative of the determinant.
If A t is continuously differentiable, then. Let p A t be the characteristic polynomial of A.
The Cayley—Hamilton theorem states that. 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 p A t.
These coefficients can be explicitly represented in terms of traces of powers of A using complete exponential Bell polynomials. The resulting formula is. The same formula follows directly from the terminating step of the Faddeev—LeVerrier algorithmmtarix efficiently determines the characteristic polynomial of A. The adjugate can be viewed in abstract terms using exterior algebras.
Let V be an n -dimensional vector space. The exterior product defines a bilinear pairing. Therefore, it yields an isomorphism. The adjugate of T is the composite. Fix a basis vector e i of R n. Consequently, its matrix representation is the adjugate of A. A vector v in R n corresponds to the linear functional. Let I c and J c denote the complements of I and Jrespectively.
Then the IJ entry of adj r A is. Iteratively taking the adjugate of an invertible matrix A k times yields. From Wikipedia, the free encyclopedia. The Theory of Matrices. Linear Algebra and its Applications 3rd ed. The Theory of Matrices in Numerical Analysis. Dover Books on Mathematics.
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