Difference between revisions of "Adjugate Formula"
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\mathbf{C}(\mathbf{A})=\left[\begin{array}{cccc} | \mathbf{C}(\mathbf{A})=\left[\begin{array}{cccc} | ||
C_{11}(\mathbf{A}) & C_{12}(\mathbf{A}) & \cdots & C_{1n}(\mathbf{A})\\ | C_{11}(\mathbf{A}) & C_{12}(\mathbf{A}) & \cdots & C_{1n}(\mathbf{A})\\ | ||
− | C_{21}(\mathbf{A}) & C_{22}(\mathbf{A}) & & C_{ | + | C_{21}(\mathbf{A}) & C_{22}(\mathbf{A}) & & C_{2n}(\mathbf{A})\\ |
\vdots & & \ddots & \vdots\\ | \vdots & & \ddots & \vdots\\ | ||
C_{n1}(\mathbf{A}) & C_{n2}(\mathbf{A}) & \cdots & C_{nn}(\mathbf{A}) | C_{n1}(\mathbf{A}) & C_{n2}(\mathbf{A}) & \cdots & C_{nn}(\mathbf{A}) |
Revision as of 15:34, 22 May 2014
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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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