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linalg::adjoint -- Adjoint of a matrix

Introduction

linalg::adjoint(A) computes the adjoint Adj(A) of the nxn matrix A. The adjoint matrix satisfies the equation A*Adj(A) = det(A)*I, where I is the n x n identity matrix.

Call(s)

linalg::adjoint(A)

Parameters

A - a square matrix of a domain of category Cat::Matrix

Returns

a matrix of the same domain type as A.

Related Functions

linalg::det

Details

Example 1

We define a matrix over the rationals:

>> MatQ := Dom::Matrix( Dom::Rational ):
   A := MatQ( [[0, 2, 1], [2, 1, 0], [1, 0, 2]] )
                               +-         -+
                               |  0, 2, 1  |
                               |           |
                               |  2, 1, 0  |
                               |           |
                               |  1, 0, 2  |
                               +-         -+

Then the adjoint matrix of A is given by:

>> Ad := linalg::adjoint(A)
                             +-            -+
                             |   2, -4, -1  |
                             |              |
                             |  -4, -1,  2  |
                             |              |
                             |  -1,  2, -4  |
                             +-            -+

We check the property of the adjoint matrix Ad mentioned above:

>> A * Ad = linalg::det(A)*MatQ::identity(3)
                    +-            -+   +-            -+
                    |  -9,  0,  0  |   |  -9,  0,  0  |
                    |              |   |              |
                    |   0, -9,  0  | = |   0, -9,  0  |
                    |              |   |              |
                    |   0,  0, -9  |   |   0,  0, -9  |
                    +-            -+   +-            -+

Background




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