Difference between revisions of "Determinant of a matrix"
(→4-by-4 matrices) |
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0 & 0 & 0 & 1 | 0 & 0 & 0 & 1 | ||
\end{array}\right]</math><br/><br/> | \end{array}\right]</math><br/><br/> | ||
− | For matrix <math>^R\mathbf{T}_N</math> it is useful to choose row 4 because it contains three zero values as factors:<br/><br/> | + | For the matrix <math>^R\mathbf{T}_N</math> it is useful to choose row 4 because it contains three zero values as factors:<br/><br/> |
<math>\begin{align} | <math>\begin{align} | ||
\det(^R\mathbf{T}_N)&= | \det(^R\mathbf{T}_N)&= | ||
\left|\begin{array}{cccc} | \left|\begin{array}{cccc} | ||
− | 1 & | + | 0 & 1 & 0 & 2a\\ |
− | + | 0 & 0 & -1 & 0\\ | |
− | \mathbf{0} & \mathbf{ | + | -1 & 0 & 0 & 0\\ |
− | + | \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} | |
\end{array}\right| & \quad & \quad & \quad\\ | \end{array}\right| & \quad & \quad & \quad\\ | ||
&= | &= |
Revision as of 13:00, 22 May 2014
← Back: Minors and cofactors | Overview: Matrices | Next: Matrix inversion → |
The determinant can be computed for an n-by-n square matrix. In the context of matrices the determinant of a matrix is a special function that assigns a scalar value to the matrix. It is denoted with or in matrix structure with vertical lines:
Considering a matrix as a linear system the determinant provides information about its solvability. If the determinant is nonzero the linear system is clearly solvable. This feature is also important for matrix inversion.
2-by-2 matrices
For a 2-by-2 matrix the determinant can easily computed as follows:
Example: Determinant of a 2-by-2 matrix
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3-by-3 matrices
For 3-by-3 matrices there is a formula called Rule of Sarrus to compute the determinant. The first two columns of the matrix are noted anew on the right of matrix. Then you can form six diagonals that describe the arithmetic operations to be performed:
The sum of the diagonal elements then forms the determinant:
The Khan Academy[1] provides a good video [2] where this formula is explained. Please watch the video for further information about the computation of the determinant of a 3-by-3 matrix.
4-by-4 matrices
One possibility to compute the determinant of a 4-by-4 matrix is a formula that uses the minors and cofactors of a matrix. First one row or column has to be choosen. The sum of the four corresponding values of the row or column multiplied by the related cofactors results in the determinant. If a column is choosen, it holds:
And if a row is choosen:
Example: determinant of a 4-by-4 matrix
This example describes the computation of the determinant of the transformation matrix that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages.
For the matrix it is useful to choose row 4 because it contains three zero values as factors: |
References
- ↑ https://www.khanacademy.org/ Khan Academy
- ↑ https://www.khanacademy.org/.../finding-the-determinant-of-a-3x3-matrix-method-1 Determinant of a 3-by-3 matrix