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Determinants

Question
CBSEENMA12034829
Wired Faculty App

Use matrix method to solve the following system of equations:
2x + 3y + 3z = 5
x – 2 y + z = – 4
3x – y – 2z = 3

Solution

The given equations are
2x + 3y + 3z = 5
x – 2 y + z = – 4
3x – y – 2z = 3
These equations can be written as
         open square brackets table row 2 cell space space space space 3 end cell cell space space space 3 end cell row 1 cell space minus 2 end cell cell space space space 1 end cell row 3 cell space minus 1 end cell cell negative 2 end cell end table close square brackets space space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row cell space space space 5 end cell row cell negative 4 end cell row cell space space space 3 end cell end table close square brackets
or     AX space equals space straight B space space space space space space space space where space straight A space equals space open square brackets table row 2 cell space space space space space space 3 end cell cell space space space space space 3 end cell row 1 cell space space minus 2 end cell cell space space space space space 1 end cell row 3 cell space space space space minus 1 end cell cell space minus 2 end cell end table close square brackets comma space space space straight X space equals space open square brackets table row straight x row straight y row straight z end table close square brackets comma space space straight B space equals space open square brackets table row cell space space space 5 end cell row cell negative 4 end cell row cell space space space 3 end cell end table close square brackets space space space space
         open vertical bar straight A close vertical bar space equals space open vertical bar table row 2 cell space space space space space space space 3 end cell cell space space space space space space space 3 end cell row 1 cell space space minus 2 end cell cell space space space space space space 1 end cell row 3 cell space space minus 1 end cell cell space space minus 2 end cell end table close vertical bar space equals space 2 left parenthesis 4 plus 1 right parenthesis minus 3 left parenthesis negative 2 minus 3 right parenthesis plus 3 left parenthesis negative 1 plus 6 right parenthesis space space space space equals space 10 plus 15 plus 15 space equals space 40 space not equal to space 0 space space space space space space space rightwards double arrow space space space straight A to the power of negative 1 end exponent space exists.     
Co-factors of the elements of first row of | A | are
  open vertical bar table row cell negative 2 end cell cell space space space space space space 1 end cell row cell negative 1 end cell cell space space minus 2 end cell end table close vertical bar comma space space space minus open vertical bar table row 1 cell space space space space space 1 end cell row 3 cell space space minus 2 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space space space space minus 2 end cell row 3 cell space space space space space minus 1 end cell end table close vertical bar
or  5, 5, 5 respectively.
Co-factors of the elements of second row of | A | are
open vertical bar table row cell space space space 3 end cell cell space space space space space space 3 end cell row cell negative 1 end cell cell space space minus 2 end cell end table close vertical bar comma space space space space open vertical bar table row 2 cell space space space space space 3 end cell row 3 cell space minus 2 end cell end table close vertical bar comma space space space minus open vertical bar table row 2 cell space space space space 3 end cell row 3 cell space minus 1 end cell end table close vertical bar
or  3, – 13, 11 respectively.
Co-factors of the elements of third row of | A | are
open vertical bar table row cell space 3 end cell cell space space space 3 end cell row cell negative 2 end cell cell space space space 1 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space space space 3 end cell row 1 cell space space space 1 end cell end table close vertical bar comma space space space open vertical bar table row 2 cell space space space space space 3 end cell row 1 cell space space minus 2 end cell end table close vertical bar
or  9, 1, – 7 respectively.
therefore space space adj. space straight A space equals space open square brackets table row 5 cell space space space space space 5 end cell cell space space space space space space 5 end cell row 3 cell space space minus 13 end cell cell space space space space space 11 end cell row 3 cell space space space space space space 1 end cell cell space space minus 7 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 5 cell space space space space 3 end cell cell space space space 9 end cell row 5 cell negative 13 end cell cell space space space 1 end cell row 5 cell space space 11 end cell cell negative 7 end cell end table close square brackets space space space space space space straight A to the power of negative 1 end exponent space equals space fraction numerator adj. space straight A over denominator open vertical bar straight A close vertical bar end fraction space equals space 1 over 40 open square brackets table row 5 cell space space space space space 3 end cell cell space space space space 9 end cell row 5 cell space minus 13 end cell cell space space space 1 end cell row 5 cell space space space 11 end cell cell negative 7 end cell end table close square brackets Now comma space space straight A space straight X space equals space straight B space space space space space rightwards double arrow space space space space straight X space equals space straight A to the power of negative 1 end exponent straight B space space space space rightwards double arrow space space space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals 1 over 40 open square brackets table row 5 cell space space space space 3 end cell cell space space space 9 end cell row 5 cell space minus 13 end cell cell space space 1 end cell row 5 cell space space space 11 end cell cell space space minus 7 end cell end table close square brackets space space open square brackets table row cell space space space space 5 end cell row cell negative 4 end cell row cell space space 3 end cell end table close square brackets rightwards double arrow space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space 1 over 40 open square brackets table row cell 25 minus 12 plus 27 end cell row cell 25 plus 52 plus 3 end cell row cell 25 minus 44 minus 21 end cell end table close square brackets space space space space space rightwards double arrow space space space space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row cell space space 1 end cell row cell space space space 2 end cell row cell negative 1 end cell end table close square brackets rightwards double arrow space space space space straight x space equals space 1 comma space space space straight y space equals space 2 comma space space space straight z space equals space minus 1

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