Calculate the Inverse of a Matrix II

Inverse of a Matrix theory

 

Inverse of a Matrix

Definition method.

If the product between two matrices is the identity matrix, then we say that the matrices are “inverse”; because by multiplying them we obtain the neutral element for the product .

Consider the inverse matrix:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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It must accomplish that: A · A-1 = I

 

Observe the following example.

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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So, we obtain the following equations system to get the inverse matrix values.

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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Determinant method.

By applying this method we say that:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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Where att (A)t is attached matrix and det (A) is the determinant of the matrix A.

The A determinant solution can be seen in the corresponding section about determinants theory.

att (A)t is obtained as follows:

Given matrix A:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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Its attached is formed by the determinings which are result by suppressing a row and column of each of one of the elements.

For example, the a32 element position would be occupied by the determining which results by eliminating the third row and second column, that is:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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and as the row + column (3+2) is odd we change the sign.

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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Then we perform its transpose:

Bioprofe |Exams with exercises about physics, chemistry and mathematics | Inverse of a Matrix

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You can download the App BioProfe READER to practice this theory with self-corrected exercises.

 

 

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