Difference between revisions of "Adjugate Formula"
From Robotics
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</math><br/><br/> | </math><br/><br/> | ||
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.<br/><br/> | 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.<br/><br/> | ||
| − | + | ||
| − | + | {{Example | |
| + | |Title=inverse of a 4-by-4 matrix | ||
| + | |Contents= | ||
| + | <br/><math>\begin{align} | ||
| + | \mathbf{A}_e &= | ||
| + | \left[\begin{array}{cccc} | ||
| + | 1 & 2 & 0 & 0\\ | ||
| + | 3 & 0 & 1 & 1\\ | ||
| + | 0 & 1 & 0 & 0\\ | ||
| + | 0 & 0 & 2 & 1 | ||
| + | \end{array}\right]\\ \\ | ||
\mathbf{C}(\mathbf{A}_e)&= | \mathbf{C}(\mathbf{A}_e)&= | ||
\left[\begin{array}{cccc} | \left[\begin{array}{cccc} | ||
| Line 47: | Line 57: | ||
\end{array}\right] | \end{array}\right] | ||
\end{align}</math> | \end{align}</math> | ||
| + | }} | ||
Revision as of 16:01, 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:
![\mathbf{C}(\mathbf{A})=\left[\begin{array}{cccc}
C_{1,1}(\mathbf{A}) & C_{1,2}(\mathbf{A}) & \cdots & C_{1,n}(\mathbf{A})\\
C_{2,1}(\mathbf{A}) & C_{2,2}(\mathbf{A}) & & C_{2,n}(\mathbf{A})\\
\vdots & & \ddots & \vdots\\
C_{n,1}(\mathbf{A}) & C_{n,2}(\mathbf{A}) & \cdots & C_{n,n}(\mathbf{A})
\end{array}\right]](/wiki/robotics/images/math/c/b/7/cb7505ec4a107c95fd31601fac888472.png)
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
|
![\begin{align}
\mathbf{A}_e &=
\left[\begin{array}{cccc}
1 & 2 & 0 & 0\\
3 & 0 & 1 & 1\\
0 & 1 & 0 & 0\\
0 & 0 & 2 & 1
\end{array}\right]\\ \\
\mathbf{C}(\mathbf{A}_e)&=
\left[\begin{array}{cccc}
1 & 0 & 3 & -6\\
0 & 0 & -1 & 2\\
-2 & 1 & -6 & 12\\
0 & 0 & 1 & -1
\end{array}\right]\\ \\
\mathbf{C}(\mathbf{A}_e)^T&=
\left[\begin{array}{cccc}
1 & 0 & -2 & 0\\
0 & 0 & 1 & 0\\
3 & -1 & -6 & 1\\
-6 & 2 & 12 & -1
\end{array}\right]=\text{adj}(\mathbf{A}_e)\\ \\
\mathbf{A}_e^{-1}&=\frac{1}{\det(\mathbf{A}_e)}\text{adj}(\mathbf{A}_e)
=\frac{1}{1}
\left[\begin{array}{cccc}
1 & 0 & -2 & 0\\
0 & 0 & 1 & 0\\
3 & -1 & -6 & 1\\
-6 & 2 & 12 & -1
\end{array}\right]
=
\left[\begin{array}{cccc}
1 & 0 & -2 & 0\\
0 & 0 & 1 & 0\\
3 & -1 & -6 & 1\\
-6 & 2 & 12 & -1
\end{array}\right]
\end{align}](/wiki/robotics/images/math/3/f/7/3f7958e0bd88c6191067286e1efaff1c.png)
