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10. Inversa d'una matriu

10.1. Inversa d'una matriu 2x2

Calculem la inversa de la matriu

     A equals open parentheses table row 2 cell negative 1 end cell row 1 3 end table close parentheses

Ho fem de 3 maneres diferents.

(Per matrius 3x3 seria similar però quedarà una mica més llarg d'operacions)

                                            

- Utilitzant el determinant i els adjunts de la transposada.

  Calcular la inversa de  A equals open parentheses table row 2 cell negative 1 end cell row 1 3 end table close parentheses

                       A to the power of negative 1 end exponent equals fraction numerator a d j u n t s open parentheses A to the power of t close parentheses over denominator open vertical bar A close vertical bar end fraction

    Determinant:  open vertical bar A close vertical bar equals 2 times 3 minus 1 times left parenthesis negative 1 right parenthesis equals 6 plus 1 equals 7

    Transposada de la matriu A:     A to the power of t equals open parentheses table row 2 1 row cell negative 1 end cell 3 end table close parentheses

    Matriu d'adjunts de  A to the power of t :

                                                            open parentheses table row 3 cell negative left parenthesis negative 1 right parenthesis end cell row cell negative 1 end cell 2 end table close parentheses 

  Per tant:  

        A to the power of negative 1 end exponent equals 1 over 7 open parentheses table row 3 1 row cell negative 1 end cell 2 end table close parentheses

                       

· Plantejant un sistema d'equacions:

    Calcular la inversa de  A equals open parentheses table row 2 cell negative 1 end cell row 1 3 end table close parentheses

  Volem una matriu X tal que A times X equals I

  És a dir:

                 open parentheses table row 2 cell negative 1 end cell row 1 3 end table close parentheses times open parentheses table row a b row c d end table close parentheses equals open parentheses table row 1 0 row 0 1 end table close parentheses

                 open parentheses table row cell 2 a minus c end cell cell 2 b minus d end cell row cell a plus 3 c end cell cell b plus 3 d end cell end table close parentheses equals open parentheses table row 1 0 row 0 1 end table close parentheses space space space space space open table attributes columnalign right end attributes row cell 2 a minus c equals 1 end cell row cell a plus 3 c equals 0 end cell end table close curly brackets space space space space rightwards double arrow space space open table attributes columnalign right end attributes row cell 6 a minus 3 c equals 3 end cell row cell a plus 3 c equals 0 end cell end table close curly brackets space space space rightwards double arrow space space 7 a equals 3 space space space rightwards double arrow space a equals 3 over 7 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 3 c equals negative a equals negative 3 over 7 space space rightwards double arrow space space c equals negative fraction numerator 3 over denominator 3 times 7 end fraction equals negative 1 over 7

                    open parentheses table row cell 2 a minus c end cell cell 2 b minus d end cell row cell a plus 3 c end cell cell b plus 3 d end cell end table close parentheses equals open parentheses table row 1 0 row 0 1 end table close parentheses space space space space space open table attributes columnalign right end attributes row cell 2 b minus d equals 0 end cell row cell b plus 3 d equals 1 end cell end table close curly brackets space space space space rightwards double arrow space space open table attributes columnalign right end attributes row cell 6 b minus 3 d equals 0 end cell row cell b plus 3 d equals 1 end cell end table close curly brackets space space space rightwards double arrow space space 7 b equals 1 space space space rightwards double arrow space b equals 1 over 7 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 3 d equals 1 minus b equals 1 minus 1 over 7 space equals 6 over 7 space rightwards double arrow space space d equals fraction numerator 6 over denominator 3 times 7 end fraction equals 2 over 7

                    Per tant: 

                    A to the power of negative 1 end exponent equals 1 over 7 open parentheses table row 3 1 row cell negative 1 end cell 2 end table close parentheses

                                   

· Pel mètode de Gauss-Jordan

    Calcular la inversa de  A equals open parentheses table row 2 cell negative 1 end cell row 1 3 end table close parentheses

  De fet és el mateix que hem fet a dalt però ho expressem en forma matricial. 

  Es tracta de plantejar la matriu ampliada formada per la matriu i la matriu identitat. És a dir:

   open parentheses table row cell table row 2 cell negative 1 end cell row 1 3 end table end cell cell left enclose space table row 1 0 row 0 1 end table end enclose end cell end table close parentheses 

   Hem de fer transformacions elementals fins que en la part esquerra ens quedi la matriu identitat: open parentheses table row 1 0 row 0 1 end table close parentheses 

   open parentheses table row cell table row 2 cell negative 1 end cell row 1 3 end table end cell cell left enclose space table row 1 0 row 0 1 end table end enclose end cell end table close parentheses space space rightwards arrow space space open parentheses table row cell table row 2 cell negative 1 end cell row 0 cell negative 7 end cell end table end cell cell left enclose space table row 1 0 row 1 cell negative 2 end cell end table end enclose end cell end table close parentheses space space rightwards arrow space space open parentheses table row cell table row cell negative 14 end cell 0 row 0 cell negative 7 end cell end table end cell cell left enclose space table row cell negative 6 end cell cell negative 2 end cell row 1 cell negative 2 end cell end table end enclose end cell end table close parentheses space space rightwards arrow space space open parentheses table row cell table row 1 0 row 0 1 end table end cell cell left enclose space table row cell 3 divided by 7 end cell cell 1 divided by 7 end cell row cell negative 1 divided by 7 end cell cell 2 divided by 7 end cell end table end enclose end cell end table close parentheses

    Per tant:      

     A to the power of negative 1 end exponent equals 1 over 7 open parentheses table row 3 1 row cell negative 1 end cell 2 end table close parentheses                                             

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