Difference between revisions of "Determinant of a matrix"

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(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 paragraph 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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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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<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/>
 
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>
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<br/><br/>
  
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  For the example matrix it is useful to choose the row 3 because it contains three zero values as factors:<br/>
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{{Example
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  <math>\begin{align}
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|Title=determinant of a 4-by-4 matrix
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|Contents=
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<br/><math>
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\mathbf{A}_e  =
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\left[\begin{array}{cccc}
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1 & 2 & 0 & 0\\
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3 & 0 & 1 & 1\\
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0 & 1 & 0 & 0\\
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0 & 0 & 2 & 1
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\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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<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
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\end{align}</math><br/>
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\end{align}</math>
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}}

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 \mathbf{A} 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:

\det(\mathbf{A})=\sum_{(i,j) \isin \text{ one row or column}}{a_{i,j}C_{i,j}}

Example: determinant of a 4-by-4 matrix


\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]

For matrix \mathbf{A}_e it is useful to choose row 3 because it contains three zero values as factors:

\begin{align} \det(\mathbf{A}_e)&= \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}

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