Difference between revisions of "Adjugate Formula"
Line 39: | Line 39: | ||
\end{array}\right]</math> | \end{array}\right]</math> | ||
<br/><br/> | <br/><br/> | ||
− | The transposed of the cofactor matrix then appears as | + | The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as |
<br/><br/><math>\mathbf{C}(\mathbf{A}_e)^T= | <br/><br/><math>\mathbf{C}(\mathbf{A}_e)^T= | ||
\left[\begin{array}{cccc} | \left[\begin{array}{cccc} | ||
Line 47: | Line 47: | ||
-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 [[Determinant_of_a_4-by-4_matrix|determinant of a 4-by-4 matrix]] and equals 1. | + | The determinant is computed in the example for a [[Determinant_of_a_4-by-4_matrix|determinant of a 4-by-4 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 16:20, 9 May 2014
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
|