Difference between revisions of "Adjugate Formula"
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|Title=inverse of a 4-by-4 matrix | |Title=inverse of a 4-by-4 matrix | ||
|Contents= | |Contents= | ||
− | <br/><math> | + | <br/><math> |
\mathbf{A}_e &= | \mathbf{A}_e &= | ||
\left[\begin{array}{cccc} | \left[\begin{array}{cccc} | ||
Line 26: | Line 26: | ||
0 & 1 & 0 & 0\\ | 0 & 1 & 0 & 0\\ | ||
0 & 0 & 2 & 1 | 0 & 0 & 2 & 1 | ||
− | \end{array}\right]\\ | + | \end{array}\right]</math> |
+ | <br/><br/> | ||
+ | The calculation of two of the cofactors of <math>\mathbf{A}_e</math> has already been described in the example for [[Minors_and_cofactors|minors and cofactors]] and the determinant is computed [[Determinant_of_a_4-by-4_matrix|here]]. | ||
+ | <math>\begin{align} | ||
\mathbf{C}(\mathbf{A}_e)&= | \mathbf{C}(\mathbf{A}_e)&= | ||
\left[\begin{array}{cccc} | \left[\begin{array}{cccc} |
Revision as of 16:05, 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
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