Difference between revisions of "Adjugate Formula"
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-6 & 2 & 12 & -1 | -6 & 2 & 12 & -1 | ||
\end{array}\right]=\text{adj}(\mathbf{A}_e)</math><br/><br/> | \end{array}\right]=\text{adj}(\mathbf{A}_e)</math><br/><br/> | ||
− | The determinant is computed in the example for a [[ | + | The determinant is computed in the example for a [[Determinant_of_a_matrix|determinant of a matrix]] and equals 1. So the inverse of <math>\mathbf{A}_e</math> is determined as <br/><br/> |
<math> | <math> | ||
\mathbf{A}_e^{-1}=\frac{1}{\det(\mathbf{A}_e)}\text{adj}(\mathbf{A}_e) | \mathbf{A}_e^{-1}=\frac{1}{\det(\mathbf{A}_e)}\text{adj}(\mathbf{A}_e) |
Revision as of 13:34, 19 May 2014
← Back: Gauß-Jordan-Algorithm | Overview: Matrix inversion | Next: Matrix inversion → |
The adjugate formula defines the inverse of an n-by-n square matrix as
where is the so called adjugate matrix of . The adjugate matrix is the transposed of the cofactor matrix:
And the cofactor matrix is just a matrix where each cell corresponds to the related cofactor:
So to determine the inverse of an n-by-n square matrix you have to compute the n square cofactors, then transpose the resulting cofactor matrix and divide all the values by the determinant.
Example: inverse of a 4-by-4 matrix using the adjugate formula
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