Difference between revisions of "Determinant of a matrix"
From Robotics
(Created page with " This paragraph describes a formula to compute the determinant of a 4-by-4 matrix using minors and cofactors of a matrix.<br/> <br/> To compute the determinant of matrix <ma...") |
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− | This | + | This article describes a formula to compute the determinant of a 4-by-4 matrix using minors and cofactors of a matrix.<br/> |
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To compute the determinant of matrix <math>\mathbf{A}</math> first one row or column is choosen. The sum of the four corresponding values of the row or column multiplied by the related cofactors results in the determinant:<br/><br/> | To compute the determinant of matrix <math>\mathbf{A}</math> first one row or column is choosen. The sum of the four corresponding values of the row or column multiplied by the related cofactors results in the determinant:<br/><br/> | ||
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\det(\mathbf{A})=\sum_{(i,j) \isin \text{ one row or column}}{a_{i,j}C_{i,j}} | \det(\mathbf{A})=\sum_{(i,j) \isin \text{ one row or column}}{a_{i,j}C_{i,j}} | ||
</math> | </math> | ||
+ | <br/><br/> | ||
− | + | {{Example | |
− | + | |Title=determinant of a 4-by-4 matrix | |
+ | |Contents= | ||
+ | <br/><math> | ||
+ | \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]</math><br/><br/> | ||
+ | For matrix <math>\mathbf{A}_e</math> it is useful to choose row 3 because it contains three zero values as factors:<br/><br/> | ||
+ | <math>\begin{align} | ||
\det(\mathbf{A}_e)&= | \det(\mathbf{A}_e)&= | ||
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
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&= 0&+&1\cdot(-1)\cdot(-1)&+&0&+&0\\ | &= 0&+&1\cdot(-1)\cdot(-1)&+&0&+&0\\ | ||
&= 1&\quad&\quad&\quad | &= 1&\quad&\quad&\quad | ||
− | \end{align}</math> | + | \end{align}</math> |
+ | }} |
Revision as of 15:46, 9 May 2014
This article describes a formula to compute the determinant of a 4-by-4 matrix using minors and cofactors of a matrix.
To compute the determinant of matrix first one row or column is choosen. The sum of the four corresponding values of the row or column multiplied by the related cofactors results in the determinant:
Example: determinant of a 4-by-4 matrix
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