Attached matrix

Attached matrix

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An adjoint matrix is ​​a linear transformation of the original matrix through the determinant of minors and its sign and is mainly used to obtain the inverse matrix.

In other words, an adjoint matrix is ​​the result of changing the sign of the determinant of each of the minors of the original matrix as a function of the position of the minor within the matrix.

The adjoint matrix of a W matrix is ​​represented as Adj (W).

The order of the original matrix and the adjoining matrix match, that is, the adjoining matrix will have the same number of columns and rows as the original matrix.

Recommended articles: main diagonal, matrix operations, square matrix.

Given any matrix W of order n we define the elements of row i and the elements of column j of W as w ij .

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Matrix of order n.

Attached matrix formula

The adjoint matrix of matrix W is obtained from:

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Attached matrix formula.

In matrices of order 2, W ij is the element w that corresponds to row i and column j. So, det (W ij ) is element w of row i and column j.

In matrices of order greater than or equal to 3, W ij is the smallest obtained by eliminating row i and column j from matrix W. Then, det (W ij ) is the determinant of the smallest W ij .

It is important to take into account the sign change that we must apply when the sum of the rows and columns with which we are working add up to an odd number. In the case that they add an even number, the negative sign will produce a neutral effect on the smaller one.

Applications

The adjoint matrix is ​​applied to obtain the inverse matrix of a matrix with nonzero determinant (0). So, to obtain the inverse matrix, we must demand that the matrix be square and invertible, that is, that it be a regular matrix. Instead, to calculate the adjoint matrix we only have to find the minors of the matrix.

Theoretical example

Order 2 matrix

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Order matrix 2.
  1. We substitute the elements of the array in the above formula.
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Procedure to obtain the adjoint matrix of a matrix of order 2.

Matrix of order 3

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Order matrix 3.
  1. We substitute the elements of the array in the above formula.
  2. We calculate the determinant of each minor.
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Procedure to obtain the adjoint matrix of a matrix of order 3.

Matrix division

  • Determinant of a matrix
  • Cholesky decomposition
  • Main diagonal