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Determinants

Question
CBSEENMA12034899
Wired Faculty App

Use matrix method to solve the following system of equations:
2x + 6y = 2
3x – z =  8
2x - y + z = – 3 

Solution

The given equations are
2x + 6y = 2
3x – z = –8
2x – y + z = –3
These equations can be written as
                   open square brackets table row 2 cell space space space space 6 end cell cell space space space 0 end cell row 3 cell space space space space space 0 end cell cell space minus 1 end cell row 2 cell space space minus 1 end cell cell space space space space space space 1 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 open square brackets table row cell space space space 2 end cell row cell negative 8 end cell row cell negative 3 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 space 6 end cell cell space space 0 end cell row 3 cell space space space 0 end cell cell space minus 1 end cell row 2 cell space minus 1 end cell cell space space 1 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 space straight B space equals space open square brackets table row cell space space 2 end cell row cell negative 8 end cell row cell space minus 3 end cell end table close square brackets
Now comma space open vertical bar straight A close vertical bar space equals open vertical bar table row 2 cell space space space 6 end cell cell space space 0 end cell row 3 cell space space 0 end cell cell negative 1 end cell row 2 cell negative 1 end cell cell space space 1 end cell end table close vertical bar space equals space 2 space open vertical bar table row cell space space 0 end cell cell space space space minus 1 end cell row cell negative 1 end cell cell space space space space space 1 end cell end table close vertical bar space minus space 6 space open vertical bar table row 3 cell space space space minus 1 end cell row 2 cell space space space space space space 1 end cell end table close vertical bar plus 0 space open vertical bar table row 3 cell space space space space space space 0 end cell row 2 cell space space minus 1 end cell end table close vertical bar space space space space space space space equals 2 space left parenthesis 0 minus 1 right parenthesis space minus space 6 left parenthesis 3 plus 2 right parenthesis space plus space 0 left parenthesis negative 3 minus 0 right parenthesis space equals space minus 2 minus 30 plus 0 space equals negative 32 space not equal to space 0 therefore space space space space straight A to the power of negative 1 end exponent space exists. straight A subscript 11 space equals space open vertical bar table row 0 cell space space space space minus 1 end cell row cell negative 1 end cell cell space space space space space space 1 end cell end table close vertical bar space equals space 0 minus 1 space equals space minus 1 straight A subscript 12 space equals space open vertical bar table row 3 cell space space space space minus 1 end cell row 2 cell space space space space space space space 1 end cell end table close vertical bar space equals space minus left parenthesis 3 plus 2 right parenthesis space equals space minus 5 straight A subscript 13 space equals space open vertical bar table row 3 cell space space space space space space 0 end cell row 2 cell space space minus 1 end cell end table close vertical bar space equals space minus 3 minus 0 space equals space minus 3 straight A subscript 21 space equals space minus open vertical bar table row 6 cell space space space space 0 end cell row cell negative 1 end cell cell space space space 1 end cell end table close vertical bar space equals space minus left parenthesis 6 plus 0 right parenthesis space equals space minus 6 straight A subscript 22 space equals space open vertical bar table row 2 cell space space space space space 0 end cell row 2 cell space space space space space 1 end cell end table close vertical bar space equals space 2 minus 0 space equals space 2 straight A subscript 23 space equals space minus open vertical bar table row 2 cell space space space space 6 end cell row 2 cell space space minus 1 end cell end table close vertical bar space equals space minus left parenthesis negative 2 minus 12 right parenthesis space equals space 14 straight A subscript 31 space equals space open vertical bar table row 6 cell space space space space space space space 0 end cell row 0 cell space space space minus 1 end cell end table close vertical bar space equals space minus 6 minus 0 space equals space minus 6 straight A subscript 32 space equals space minus open vertical bar table row 2 cell space space space space space space space space 0 end cell row 3 cell space space space space minus 1 end cell end table close vertical bar space equals space minus left parenthesis negative 2 minus 0 right parenthesis space equals space 2 
straight A subscript 33 space equals space open vertical bar table row 2 cell space space space 6 end cell row 3 cell space space 0 end cell end table close vertical bar space equals space 0 minus 8 space equals space minus 18
   adj space straight A space equals space open square brackets table row cell negative 1 end cell cell space space space minus 5 end cell cell space space space minus 3 end cell row cell negative 6 end cell cell space space space space space 2 end cell cell space space space space space 14 end cell row cell negative 6 end cell cell space space space space 2 end cell cell space space minus 18 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 1 end cell cell space space minus 6 end cell cell space space minus 6 end cell row cell negative 5 end cell cell space space space space 2 end cell cell space space space space 2 end cell row cell negative 3 end cell cell space space 14 end cell cell space minus 18 end cell end table close square brackets 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 minus 1 over 32 open square brackets table row cell negative 1 end cell cell space space minus 6 end cell cell space minus 6 end cell row cell negative 5 end cell cell space space space 2 end cell cell space space space space space 2 end cell row cell negative 3 end cell cell space space 14 end cell cell space minus 18 end cell end table close square brackets Now comma space space space AX space equals space straight B space space space space space space 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 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 minus 1 over 32 open square brackets table row cell negative 1 end cell cell space space space minus 6 end cell cell space space space minus 6 end cell row cell negative 5 end cell cell space space space space space 2 end cell cell space space space space space space 2 end cell row cell negative 3 end cell cell space space space space 14 end cell cell space minus 18 end cell end table close square brackets space space open square brackets table row cell space space space space 2 end cell row cell negative 8 end cell row cell negative 3 end cell end table close square brackets space space space space space 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 minus 1 over 32 open square brackets table row cell negative 2 plus 48 plus 18 end cell row cell negative 10 minus 16 minus 6 end cell row cell negative 6 minus 112 plus 54 end cell end table close square brackets 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 minus 1 over 32 open square brackets table row cell space space space 64 end cell row cell negative 32 end cell row cell negative 64 end cell end table close square brackets 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 space equals space open square brackets table row cell negative 2 end cell row cell space 1 end cell row cell space space 2 end cell 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 space straight y space equals space 1 comma space space space space space straight z space equals space 2.

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