Difference between revisions of "Determinant of a matrix"

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|Title=determinant of a 4-by-4 matrix
 
|Title=determinant of a 4-by-4 matrix
 
|Contents=
 
|Contents=
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This example describes the computation of the determinant of the transformation matrix <math>^RT_N</math> introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages.
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This example describes the computation of the determinant of the transformation matrix <math>^R\mathbf{T}_N</math> that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages.
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<br/><math>
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<br/><br/>
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\mathbf{A}_e =  
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:<math>
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^R\mathbf{T}_N =  
 
\left[\begin{array}{cccc}
 
\left[\begin{array}{cccc}
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1 & 2 & 0 & 0\\
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0 & 1 & 0 & 2a\\
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3 & 0 & 1 & 1\\
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0 & 0 & -1 & 0\\
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0 & 1 & 0 & 0\\
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-1 & 0 & 0 & 0\\
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0 & 0 & 2 & 1
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0 & 0 & 0 & 1
 
\end{array}\right]</math><br/><br/>
 
\end{array}\right]</math><br/><br/>
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For matrix <math>\mathbf{A}_e</math> it is useful to choose row 3 because it contains three zero values as factors:<br/><br/>
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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/>
 
<math>\begin{align}
 
<math>\begin{align}
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\det(\mathbf{A}_e)&=
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\det(^R\mathbf{T}_N)&=
 
\left|\begin{array}{cccc}
 
\left|\begin{array}{cccc}
 
1 & 2 & 0 & 0\\
 
1 & 2 & 0 & 0\\

Revision as of 11:30, 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 \det(\mathbf{A}) or in matrix structure with vertical lines:

\det(\mathbf{A})= \begin{array}{|ccc|} a_{11} & \dots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{n1} & \dots & a_{nn} \end{array}

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:

\det(\mathbf{A})= \left|\begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array}\right|=a_{11}a_{22}-a_{21}a_{12}


Example: Determinant of a 2-by-2 matrix


\det \left[\begin{array}{cc} 1 & 3\\ 2 & 2 \end{array}\right] = \left|\begin{array}{cc} 1 & 3\\ 2 & 2 \end{array}\right|= 1\cdot 2-3\cdot 2= 2-6=-4


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:

Sarrus.png

The sum of the diagonal elements then forms the determinant:

\det \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix} = \begin{vmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} - a_{13} a_{22} a_{31} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33}

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 j is choosen, it holds:

\det(\mathbf{A})=\sum_{i=1}^n a_{ij}\cdot C_{ij} \quad \text{for the choosen column } j

And if a row i is choosen:

\det(\mathbf{A})=\sum_{j=1}^n a_{ij}\cdot C_{ij} \quad \text{for the choosen row } i



Example: determinant of a 4-by-4 matrix

This example describes the computation of the determinant of the transformation matrix ^R\mathbf{T}_N that is introduced in the robotics script in chapter 3 on page 3-37 and used on the following pages.

^R\mathbf{T}_N = \left[\begin{array}{cccc} 0 & 1 & 0 & 2a\\ 0 & 0 & -1 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array}\right]

For matrix ^R\mathbf{T}_N it is useful to choose row 4 because it contains three zero values as factors:

\begin{align} \det(^R\mathbf{T}_N)&= \left|\begin{array}{cccc} 1 & 2 & 0 & 0\\ 3 & 0 & 1 & 1\\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0}\\ 0 & 0 & 2 & 1 \end{array}\right| & \quad & \quad & \quad\\ &= \mathbf{0}\cdot C_{3,1}&+&\mathbf{1}\cdot C_{3,2}&+&\mathbf{0}\cdot C_{3,3}&+&\mathbf{0}\cdot C_{3,4}\\ &= 0\cdot(-1)^{3+1} \left|\begin{array}{ccc} 2 & 0 & 0\\ 0 & 1 & 1\\ 0 & 2 & 1 \end{array}\right| &+&1\cdot(-1)^{3+2} \left|\begin{array}{ccc} 1 & 0 & 0\\ 3 & 1 & 1\\ 0 & 2 & 1 \end{array}\right| &+&0\cdot(-1)^{3+3} \left|\begin{array}{ccc} 1 & 2 & 0\\ 3 & 0 & 1\\ 0 & 0 & 1 \end{array}\right| &+&0\cdot(-1)^{3+4} \left|\begin{array}{ccc} 1 & 2 & 0\\ 3 & 0 & 1\\ 0 & 0 & 2 \end{array}\right|\\ &= 0&+&1\cdot(-1)\cdot(-1)&+&0&+&0\\ &= 1&\quad&\quad&\quad \end{align}


References

  1. ↑ https://www.khanacademy.org/ Khan Academy
  2. ↑ https://www.khanacademy.org/.../finding-the-determinant-of-a-3x3-matrix-method-1 Determinant of a 3-by-3 matrix
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