Difference between revisions of "Manuals/calci/DET"
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| − | ==DET== | + | <div style="font-size:30px">'''DET(array)'''</div><br/> |
| + | *<math>array</math> is the set of numbers. | ||
| + | |||
| + | ==Description== | ||
| + | *This function gives the determinant value of a matrix. | ||
| + | *To calculate the determinant of a matrix, we can choose only square matrix.i.e. Number of rows and number of columns should be equal. | ||
| + | *Determinant of the identity matrix is always 1. | ||
| + | *Determinant of the matrix <math>A</math> is denoted by <math>det(A)</math> or <math>|A|</math>. | ||
| + | *Let <math>A</math> be 2x2 matrix with the elements | ||
| + | <math>A = \begin{bmatrix} | ||
| + | a & b \\ | ||
| + | c & d \\ | ||
| + | \end{bmatrix} | ||
| + | </math> | ||
| + | *Then <math>det(A)=ad-bc</math>, where <math>a,b,c,d</math> all are real numbers. | ||
| + | *Let <math>A</math> be the 3x3 matrix with the elements | ||
| + | <math>A = \begin{bmatrix} | ||
| + | a & b & c \\ | ||
| + | d & e & f \\ | ||
| + | g & h & i \\ | ||
| + | \end{bmatrix} | ||
| + | </math> | ||
| + | Then <math>|A|=a\begin{vmatrix} | ||
| + | e & f \\ | ||
| + | h & i | ||
| + | \end{vmatrix} -b\begin{vmatrix} | ||
| + | d & f \\ | ||
| + | g & i | ||
| + | \end{vmatrix} +c\begin{vmatrix} | ||
| + | d & e \\ | ||
| + | g & h | ||
| + | \end{vmatrix}</math>: | ||
| + | <math>|A| =a(ei-fh)-b(di-fg)+c(dh-eg)</math> | ||
| + | *Let <math>A</math> be a square matrix of order <math>n</math>. Write <math>A = (a_{ij})</math>, | ||
| + | *Where <math>a_{ij}</math> is the entry on the <math>i^{th}</math> row and <math>j^{th}</math> column and <math>i=1</math> to <math>n</math> & <math>j=1</math> to <math>n</math>. | ||
| + | *For any <math>i</math> and <math>j</math>, set <math>A_{ij}</math> (called the co-factors), then the general formula for determinant of the matrix <math>A</math> is, | ||
| + | <math>|A|=\sum_{j=1}^n a_{ij} A_{ij}</math>, for any fixed <math>i</math>. | ||
| + | Also<math>|A|=\sum_{i=1}^n a_{ij} A_{ij}</math>, for any fixed <math>j</math>. | ||
| + | *This function will give the result as error when | ||
| + | 1. Any one of the element in array is empty or contain non-numeric | ||
| + | 2. Number of rows is not equal to number of columns | ||
| + | |||
| + | |||
| + | ==Examples== | ||
| + | #=DET([[6,4,8],[3,6,1],[2,4,5]]) = 104 | ||
| + | #=DET([[-5,10],[6,-8]]) = -20 | ||
| + | #=DET([[1,0,2,1],[4,0,2,-1],[1,4,5,2],[3,1,2,0]]) = 17 | ||
| + | #=DET([1,2,3],[5,2,8]) = NAN | ||
| + | |||
| + | |||
| + | ==Related Videos== | ||
| + | |||
| + | {{#ev:youtube|v=H9BWRYJNIv4|280|center|Determinants}} | ||
| + | |||
| + | |||
| + | ==See Also== | ||
| + | *[[Manuals/calci/MINVERSE | MINVERSE ]] | ||
| + | *[[Manuals/calci/MMULT | MMULT ]] | ||
| + | |||
| + | ==References== | ||
| + | *[http://en.wikipedia.org/wiki/Determinant Determinant ] | ||
| + | |||
| + | *[[Z_API_Functions | List of Main Z Functions]] | ||
| + | *[[ Z3 | Z3 home ]] | ||
Latest revision as of 04:43, 26 May 2020
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle array} is the set of numbers.
Description
- This function gives the determinant value of a matrix.
- To calculate the determinant of a matrix, we can choose only square matrix.i.e. Number of rows and number of columns should be equal.
- Determinant of the identity matrix is always 1.
- Determinant of the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is denoted by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle det(A)} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A|} .
- Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} be 2x2 matrix with the elements
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}}
- Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle det(A)=ad-bc} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c,d} all are real numbers.
- Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} be the 3x3 matrix with the elements
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} } Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A|=a\begin{vmatrix} e & f \\ h & i \end{vmatrix} -b\begin{vmatrix} d & f \\ g & i \end{vmatrix} +c\begin{vmatrix} d & e \\ g & h \end{vmatrix}} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A| =a(ei-fh)-b(di-fg)+c(dh-eg)}
- Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} be a square matrix of order . Write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = (a_{ij})} ,
- Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ij}} is the entry on the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} row and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{th}} column and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} & Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=1} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .
- For any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{ij}} (called the co-factors), then the general formula for determinant of the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A|=\sum_{j=1}^n a_{ij} A_{ij}} , for any fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} . AlsoFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A|=\sum_{i=1}^n a_{ij} A_{ij}} , for any fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} .
- This function will give the result as error when
1. Any one of the element in array is empty or contain non-numeric 2. Number of rows is not equal to number of columns
Examples
- =DET([[6,4,8],[3,6,1],[2,4,5]]) = 104
- =DET([[-5,10],[6,-8]]) = -20
- =DET([[1,0,2,1],[4,0,2,-1],[1,4,5,2],[3,1,2,0]]) = 17
- =DET([1,2,3],[5,2,8]) = NAN
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