Difference between revisions of "Manuals/calci/ADJ"
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(Created page with "<div style="font-size:30px">'''ADJ(Array)'''</div><br/> *<math>Array</math> is the set of values. ==Description== *This function shows the Adjoint of a given matrix. *In <mat...") |
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*Consider3x3 matrix <math>A=\begin{pmatrix} | *Consider3x3 matrix <math>A=\begin{pmatrix} | ||
a_{11} & a_{12} & a_{13} \\ | a_{11} & a_{12} & a_{13} \\ | ||
− | + | a_{21} & a_{22} & a_{23} \\ | |
− | \end{pmatrix} </math> | + | a_{31} & a_{32} & a_{33} |
− | + | \end{pmatrix} </math>. | |
+ | *Its adjugate is the transpose of its cofactor matrix:<math>adj(A)=C^{T} = | ||
+ | \begin{pmatrix} | ||
+ | +\begin{vmatrix} | ||
+ | a_ {22}& a_{23} \\ | ||
+ | a_ {32}& a_{33} | ||
+ | \end{vmatrix} & - \begin{vmatrix} | ||
+ | a_ {12}& a_{13} \\ | ||
+ | a_ {32}& a_{33} | ||
+ | \end{vmatrix} & +\begin{vmatrix} | ||
+ | a_ {12}& a_{13} \\ | ||
+ | a_ {22}& a_{23} | ||
+ | \end{vmatrix} \\ | ||
+ | +\begin{vmatrix} | ||
+ | a_ {21}& a_{23} \\ | ||
+ | a_ {31}& a_{33} | ||
+ | \end{vmatrix} & - \begin{vmatrix} | ||
+ | a_ {11}& a_{13} \\ | ||
+ | a_ {31}& a_{33} | ||
+ | \end{vmatrix} & +\begin{vmatrix} | ||
+ | a_ {11}& a_{13} \\ | ||
+ | a_ {21}& a_{23} | ||
+ | \end{vmatrix} \\ | ||
+ | +\begin{vmatrix} | ||
+ | a_ {21}& a_{22} \\ | ||
+ | a_ {31}& a_{32} | ||
+ | \end{vmatrix} & - \begin{vmatrix} | ||
+ | a_ {11}& a_{12} \\ | ||
+ | a_ {31}& a_{32} | ||
+ | \end{vmatrix} & +\begin{vmatrix} | ||
+ | a_ {11}& a_{12} \\ | ||
+ | a_ {21}& a_{22} | ||
+ | \end{vmatrix} \\ | ||
+ | \end{pmatrix}</math> | ||
+ | ==Examples== | ||
+ | 1.adj([[10,12],[-14,21]]) | ||
+ | {| class="wikitable" | ||
+ | |- | ||
+ | | 21 ||-12 | ||
+ | |- | ||
+ | |14 || 10 | ||
+ | |} | ||
+ | 2.adj([[4,5,8],[3,-6,-9],[10,-12,4]]) | ||
+ | {| class="wikitable" | ||
+ | |- | ||
+ | | -132 ||-116||2.9999999999999982 | ||
+ | |- | ||
+ | | -102|| -64|| 60 | ||
+ | |- | ||
+ | | 24 || 98 || -39 | ||
+ | |} | ||
+ | |||
+ | 3.adj([[5,-2,2,7],[1,0,0,3], [-3,1,5,0], [3,-1,-9,4]]) | ||
+ | {| class="wikitable" | ||
+ | |- | ||
+ | | -12 ||76||-60 ||-36 | ||
+ | |- | ||
+ | | -56 || 207.99999999999997 || -81.99999999999999 || -57.99999999999999 || | ||
+ | |- | ||
+ | | 4 || 3.999999999999999 || -1.9999999999999998 || -10 || | ||
+ | |- | ||
+ | | 4 ||3.9999999999999982 || 20 || 12 || | ||
+ | |} | ||
+ | |||
+ | ==Related Videos== | ||
+ | |||
+ | {{#ev:youtube|v=oHzpMgKuI9Q|280|center|Adjoint Matrix}} | ||
+ | |||
+ | ==See Also== | ||
+ | *[[Manuals/calci/MATRIXADJOINT | MATRIXADJOINT ]] | ||
+ | *[[Manuals/calci/MINVERSE | MINVERSE ]] | ||
+ | *[[Manuals/calci/MMULT | MMULT ]] | ||
==References== | ==References== | ||
*[https://en.wikipedia.org/wiki/Adjugate_matrix Adjugate matrix] | *[https://en.wikipedia.org/wiki/Adjugate_matrix Adjugate matrix] | ||
+ | |||
+ | *[[Z_API_Functions | List of Main Z Functions]] | ||
+ | |||
+ | *[[ Z3 | Z3 home ]] |
Latest revision as of 14:24, 9 April 2019
ADJ(Array)
- is the set of values.
Description
- This function shows the Adjoint of a given matrix.
- In , is the set of matrix values.
- Adjoint of a matrix is called adjugate, classical adjoint, or adjunct.Adjoint of a matrix formed by taking the transpose of the cofactor matrix of a given original Square matrix.
- Adjoint of matrix A is written by .
- The adjugate of A is the transpose of the cofactor matrix C of A, .
- Also adjoint of a matrix is defined by .
- The adjugate of 1x1 matrix is .
- The adjugate of 2x2 matrix is .
- Consider3x3 matrix .
- Its adjugate is the transpose of its cofactor matrix:
Examples
1.adj([[10,12],[-14,21]])
21 | -12 |
14 | 10 |
2.adj([[4,5,8],[3,-6,-9],[10,-12,4]])
-132 | -116 | 2.9999999999999982 |
-102 | -64 | 60 |
24 | 98 | -39 |
3.adj([[5,-2,2,7],[1,0,0,3], [-3,1,5,0], [3,-1,-9,4]])
-12 | 76 | -60 | -36 | |
-56 | 207.99999999999997 | -81.99999999999999 | -57.99999999999999 | |
4 | 3.999999999999999 | -1.9999999999999998 | -10 | |
4 | 3.9999999999999982 | 20 | 12 |