find the matrix X for which 
     3  2   .X. -1  1  =   2  -1
     7  5        -2  1       0   4

Asked by science field | 11th May, 2015, 02:40: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space e q u a t i o n open square brackets table row 3 2 row 7 5 end table close square brackets times X times open square brackets table row cell negative 1 end cell 1 row cell negative 2 end cell 1 end table close square brackets equals open square brackets table row 2 cell negative 1 end cell row 0 4 end table close square brackets L e t space u s space w r i t e space t h e space a b o v e space e q u a t i o n space a s A cross times X cross times B equals C comma space w h e r e space A equals open square brackets table row 3 2 row 7 5 end table close square brackets comma space B equals open square brackets table row cell negative 1 end cell 1 row cell negative 2 end cell 1 end table close square brackets space a n d space C equals open square brackets table row 2 cell negative 1 end cell row 0 4 end table close square brackets N o w space p r e space m u l t i p l y space A to the power of negative 1 end exponent space a n d space p o s t space m u l t i p l y space B to the power of negative 1 end exponent space t o space t h e space a b o v e space e q u a t i o n comma space w e space h a v e comma A to the power of negative 1 end exponent cross times A cross times X cross times B cross times B to the power of negative 1 end exponent equals A to the power of negative 1 end exponent cross times C cross times B to the power of negative 1 end exponent rightwards double arrow I cross times X cross times I equals A to the power of negative 1 end exponent cross times C cross times B to the power of negative 1 end exponent rightwards double arrow X equals A to the power of negative 1 end exponent cross times C cross times B to the power of negative 1 end exponent N o w space c o n s i d e r space f i n d i n g space t h e space i n v e r s e space o f space open square brackets table row 3 2 row 7 5 end table close square brackets A equals open square brackets table row 3 2 row 7 5 end table close square brackets therefore open vertical bar A close vertical bar equals open vertical bar table row 3 2 row 7 5 end table close vertical bar equals 15 minus 14 equals 1 not equal to 0 S o space A to the power of negative 1 end exponent space i s space i n v e r t i b l e therefore A to the power of negative 1 end exponent equals fraction numerator 1 over denominator open vertical bar A close vertical bar end fraction a d j A equals 1 over 1 open square brackets table row 5 cell negative 2 end cell row cell negative 7 end cell 3 end table close square brackets rightwards double arrow A to the power of negative 1 end exponent equals open square brackets table row 5 cell negative 2 end cell row cell negative 7 end cell 3 end table close square brackets N o w space c o n s i d e r space f i n d i n g space t h e space i n v e r s e space o f space open square brackets table row cell negative 1 end cell 1 row cell negative 2 end cell 1 end table close square brackets B equals open square brackets table row cell negative 1 end cell 1 row cell negative 2 end cell 1 end table close square brackets therefore open vertical bar B close vertical bar equals open vertical bar table row cell negative 1 end cell 1 row cell negative 2 end cell 1 end table close vertical bar equals negative 1 minus open parentheses negative 2 close parentheses equals 1 not equal to 0 S o space B to the power of negative 1 end exponent space i s space i n v e r t i b l e therefore B to the power of negative 1 end exponent equals fraction numerator 1 over denominator open vertical bar B close vertical bar end fraction a d j B equals 1 over 1 open square brackets table row 1 cell negative 1 end cell row 2 cell negative 1 end cell end table close square brackets rightwards double arrow B to the power of negative 1 end exponent equals open square brackets table row 1 cell negative 1 end cell row 2 cell negative 1 end cell end table close square brackets S i n c e space X equals A to the power of negative 1 end exponent cross times C cross times B comma space w e space h a v e comma X equals open square brackets table row 5 cell negative 2 end cell row cell negative 7 end cell 3 end table close square brackets cross times open square brackets table row 2 cell negative 1 end cell row 0 4 end table close square brackets cross times open square brackets table row 1 cell negative 1 end cell row 2 cell negative 1 end cell end table close square brackets rightwards double arrow X equals open square brackets table row cell 5 cross times 2 minus 2 cross times 0 end cell cell 5 cross times open parentheses negative 1 close parentheses minus 2 cross times 4 end cell row cell negative 7 cross times 2 plus 3 cross times 0 end cell cell open parentheses negative 7 close parentheses cross times open parentheses negative 1 close parentheses plus 3 cross times 4 end cell end table close square brackets cross times open square brackets table row 1 cell negative 1 end cell row 2 cell negative 1 end cell end table close square brackets rightwards double arrow X equals open square brackets table row 10 cell negative 13 end cell row cell negative 14 end cell 19 end table close square brackets cross times open square brackets table row 1 cell negative 1 end cell row 2 cell negative 1 end cell end table close square brackets rightwards double arrow X equals open square brackets table row cell 10 cross times 1 minus 13 cross times 2 end cell cell 10 cross times open parentheses negative 1 close parentheses plus open parentheses negative 13 close parentheses cross times open parentheses negative 1 close parentheses end cell row cell negative 14 cross times 1 plus 19 cross times 2 end cell cell open parentheses negative 14 close parentheses cross times open parentheses negative 1 close parentheses plus 19 cross times open parentheses negative 1 close parentheses end cell end table close square brackets rightwards double arrow X equals open square brackets table row cell negative 16 end cell 3 row 24 cell negative 5 end cell end table close square brackets

Answered by Vimala Ramamurthy | 11th May, 2015, 04:46: PM

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