Difference between revisions of "Adjugate Formula"
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2&-6&-1\\ | 2&-6&-1\\ | ||
0&1&0\\ | 0&1&0\\ | ||
| − | -1&3 | + | -1&3&1 |
\end{array}\right]</math> | \end{array}\right]</math> | ||
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=\frac{1}{1} | =\frac{1}{1} | ||
\left[\begin{array}{ccc} | \left[\begin{array}{ccc} | ||
| − | 2& | + | 2&0&-1\\ |
| − | + | -6&1&3\\ | |
| − | -1& | + | -1&0&1 |
\end{array}\right] | \end{array}\right] | ||
= | = | ||
\left[\begin{array}{ccc} | \left[\begin{array}{ccc} | ||
| − | 2& | + | 2&0&-1\\ |
| − | + | -6&1&3\\ | |
| − | -1& | + | -1&0&1 |
\end{array}\right] | \end{array}\right] | ||
</math> | </math> | ||
Revision as of 16:28, 22 May 2014
| ← Back: Gauß-Jordan-Algorithm | Overview: Matrix inversion | Next: Vector algebra → |
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_{11}(\mathbf{A}) & C_{12}(\mathbf{A}) & \cdots & C_{1n}(\mathbf{A})\\
C_{21}(\mathbf{A}) & C_{22}(\mathbf{A}) & & C_{2n}(\mathbf{A})\\
\vdots & & \ddots & \vdots\\
C_{n1}(\mathbf{A}) & C_{n2}(\mathbf{A}) & \cdots & C_{nn}(\mathbf{A})
\end{array}\right]](/wiki/robotics/images/math/8/4/d/84de09df69b63883c3544929bdfaad78.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 3-by-3 matrix using the adjugate formula
The following matrix is already used in examples for the minors and cofactors and for the determinant.
The calculation of two of the cofactors of The transposed of the cofactor matrix which corresponds to the adjugate matrix then appears as
The determinant is computed in the example for a determinant of a matrix and equals 1. So the inverse of |
![\mathbf{A}_e =
\left[\begin{array}{ccc}
1&0&1\\
3&1&0\\
1&0&2
\end{array}\right]](/wiki/robotics/images/math/7/f/b/7fbcc9a947f1cf1337db334d9c371547.png)
has already been described in the example for ![\mathbf{C}(\mathbf{A}_e)=
\left[\begin{array}{ccc}
2&-6&-1\\
0&1&0\\
-1&3&1
\end{array}\right]](/wiki/robotics/images/math/0/e/8/0e83faeb4172b28867632aa6107bd448.png)
![\mathbf{C}(\mathbf{A}_e)^T=
\left[\begin{array}{ccc}
2&0&-1\\
-6&1&3\\
-1&0&1
\end{array}\right]=\text{adj}(\mathbf{A}_e)](/wiki/robotics/images/math/a/8/d/a8d568ef0360bbe1b422066124f70dcb.png)
![\mathbf{A}_e^{-1}=\frac{1}{\det(\mathbf{A}_e)}\text{adj}(\mathbf{A}_e)
=\frac{1}{1}
\left[\begin{array}{ccc}
2&0&-1\\
-6&1&3\\
-1&0&1
\end{array}\right]
=
\left[\begin{array}{ccc}
2&0&-1\\
-6&1&3\\
-1&0&1
\end{array}\right]](/wiki/robotics/images/math/7/7/3/7737967cd6fc8472ce52d1f0cc942f16.png)
| Example: inverse of a 4-by-4 matrix using the adjugate formula
|
![\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]](/wiki/robotics/images/math/a/3/8/a386cb4eac26aca26ac89fd336061a0e.png)
![\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]](/wiki/robotics/images/math/2/c/1/2c171afbf8b3bb79274d1b6d160574ad.png)
![\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)](/wiki/robotics/images/math/4/d/6/4d60f99b73fcd16a1e7f14e381e93148.png)
![\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]](/wiki/robotics/images/math/1/f/3/1f3b7de9ef6df9ca9270bf2b7d321cb6.png)
