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Determinants

Question
CBSEENMA12034800
Wired Faculty App

Solve the following system of equations by matrix method:
2x – y + z = – 3
3 x – z = – 8
2x + 6y    = 2

Solution

The given equations are
2x – y + z = –3    i.e.    2x – y + z = – 3
3x – z = –8          i.e.    3x + 0y – z = –8
2x + 6y = 2          i.e.    2x + 6 y + 0z = 2
These equations can be written as
                 open square brackets table row 2 cell space space minus 1 end cell cell space space space space space space 1 end cell row 3 cell space space space space 0 end cell cell space minus 1 end cell row 2 cell space space space space 6 end cell cell space space space space 0 end cell end table close square brackets 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 negative 3 end cell row cell negative 8 end cell row cell space 2 end cell end table close square brackets
or             AX space equals space straight B space where space straight A space equals space open square brackets table row 2 cell space space minus 1 end cell cell space space space space space 1 end cell row 3 cell space space space space 0 end cell cell space minus 1 end cell row 2 cell space space space space 6 end cell cell space space space 0 end cell end table close square brackets comma space space space straight X equals space open square brackets table row straight x row straight y row straight z end table close square brackets comma space space space straight B space equals space open square brackets table row cell negative 3 end cell row cell negative 8 end cell row cell space 2 end cell end table close square brackets
Now comma space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 2 cell space space minus 1 end cell cell space space space space 1 end cell row 3 cell space space space space 0 end cell cell negative 1 end cell row 2 cell space space space 6 end cell cell space space 0 end cell end table close vertical bar space equals space 2 open vertical bar table row 0 cell space space minus 1 end cell row 6 cell space space space space 0 end cell end table close vertical bar minus left parenthesis negative 1 right parenthesis space open vertical bar table row 3 cell space space minus 1 end cell row 2 cell space space space space 0 end cell end table close vertical bar plus 1 space open vertical bar table row 3 cell space space 0 end cell row 2 cell space space 6 end cell end table close vertical bar space space space space space space space space space space space space space space space space equals 2 left parenthesis 0 plus 6 right parenthesis space plus space 1 space left parenthesis 0 plus 2 right parenthesis space plus space 1 space left parenthesis 18 minus 0 right parenthesis space equals space 12 plus 2 plus 18 space equals space 32 space not equal to space 0 therefore space space space straight A to the power of negative 1 end exponent space exists. space space space space space space straight A subscript 11 space equals space open vertical bar table row 0 cell space space minus 1 end cell row 6 cell space space space space space 0 end cell end table close vertical bar space equals space 0 plus 6 space equals space 6 space space space space space space space straight A subscript 12 space equals space minus open vertical bar table row 3 cell space space space minus 1 end cell row 2 cell space space space space space space 0 end cell end table close vertical bar space equals space minus left parenthesis 0 plus 2 right parenthesis space equals space minus 2 space space space space space space space straight A subscript 13 space equals space open vertical bar table row 3 cell space space space 0 end cell row 2 cell space space space 6 end cell end table close vertical bar space equals space 18 minus 0 space equals space 18 space space space space space space straight A subscript 21 space equals space minus open vertical bar table row cell negative 1 end cell cell space space space 1 end cell row 6 cell space space 0 end cell end table close vertical bar space equals space minus left parenthesis 0 minus 6 right parenthesis space space equals space 6 space space space space space straight A subscript 22 space equals space open vertical bar table row 2 cell space space space space space space space 1 space space end cell row 2 cell space space space space 0 end cell end table close vertical bar space equals 0 minus 2 space equals space minus 2 space space space space space straight A subscript 23 space equals space minus open vertical bar table row 2 cell space space space minus 1 end cell row 2 cell space space space space 6 end cell end table close vertical bar space equals negative space left parenthesis 12 plus 12 right parenthesis space equals space minus 14
straight A subscript 31 space equals space open vertical bar table row cell negative 1 end cell cell space space space space 1 end cell row 0 cell negative 1 end cell end table close vertical bar space equals space 1 minus 0 space space equals 1 straight A subscript 32 space equals space minus open vertical bar table row 2 cell space space space space space space 1 end cell row 3 cell space space minus 1 end cell end table close vertical bar space equals space minus left parenthesis negative 2 minus 3 right parenthesis space equals space 5 straight A subscript 33 space equals space open vertical bar table row 2 cell space space space minus 1 end cell row 3 cell space space space space space 0 end cell end table close vertical bar space equals space 0 plus 3 space equals space 3
therefore space space space adj. space straight A space equals space open square brackets table row 6 cell space space minus 2 end cell cell space space space space space 18 end cell row 6 cell space space minus 2 end cell cell space minus 14 end cell row 1 cell space space 5 end cell cell space space space space space 3 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell space 6 end cell cell space space space space 6 end cell cell space space space 1 end cell row cell negative 2 end cell cell space minus 2 end cell cell space space space 5 end cell row 18 cell negative 14 end cell cell space space space 3 end cell end table close square brackets space space space 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 32 open square brackets table row 6 cell space space space 6 end cell cell space space 1 end cell row cell negative 2 end cell cell space minus 2 end cell cell space space 5 end cell row 18 cell space minus 14 end cell cell space space 3 end cell end table close square brackets Now comma space space space space space space space AX space equals space straight B space space space space space space space space rightwards double arrow space space space straight X space equals space straight A to the power of negative 1 end exponent straight B space space space space space space space space space rightwards double arrow space 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 1 over 32 open square brackets table row cell space space space 6 end cell cell space space space space 6 end cell cell space space space space 1 end cell row cell negative 2 end cell cell space space minus 2 end cell cell space space space 5 end cell row 18 cell space space space minus 14 end cell cell space space 3 end cell end table close square brackets space space open square brackets table row cell negative 3 end cell row cell negative 8 end cell row cell space 2 end cell end table close square brackets space rightwards double arrow space 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 32 open square brackets table row cell negative 18 minus 48 plus 2 end cell row cell 6 plus 16 plus 10 end cell row cell negative 54 plus 112 plus 6 end cell end table close square brackets space space space 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 1 over 32 open square brackets table row cell negative 64 end cell row 32 row 64 end table close square brackets space space space space space space space rightwards double arrow space 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 space open square brackets table row cell negative 2 end cell row cell space 1 end cell row 2 end table close square brackets therefore space space space space space space space space space space straight x space equals space minus 2 comma space space space space space space straight y space equals 1 comma space space space space straight z space equals space 2
    

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