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Determinants

Question
CBSEENMA12034859
Wired Faculty App

Use matrix method to solve the following system of equations:
2x – y - z = 7      
3x + y – z = 7            
x + y – z = 3

Solution

The given equations are
2x – y – z = 7    ...(1)
3x + y – z = 7    ...(2)
x + y – z = 3    ...(3)
These equations can be written as
                     open square brackets table row 2 cell space space space minus 1 end cell cell space space space minus 1 end cell row 3 cell space space space space space space space 1 end cell cell space space space space minus 1 end cell row 1 cell space space space space space space 1 end cell cell space space space minus 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 space open square brackets table row 7 row 7 row 3 end table close square brackets
or  AX space equals space straight B space space where space straight A space equals space open square brackets table row 2 cell space space minus 1 end cell cell space space minus 1 end cell row 3 cell space space space space 1 end cell cell negative 1 end cell row 1 cell space space space 1 end cell cell negative 1 end cell end table close square brackets comma space 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 space space straight B space equals space open square brackets table row 7 row 7 row 3 end table close square brackets
                        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 minus 1 end cell row 3 cell space space space space space 1 end cell cell space space space minus 1 end cell row 1 cell space space space space 1 end cell cell space space space minus 1 end cell end table close vertical bar equals space 2 space open vertical bar table row 1 cell space space space minus 1 end cell row 1 cell space space minus 1 end cell end table close vertical bar minus left parenthesis negative 1 right parenthesis space open vertical bar table row 3 cell space space space minus 1 end cell row 1 cell space space minus 1 end cell end table close vertical bar plus left parenthesis negative 1 right parenthesis space open vertical bar table row 3 cell space space space 1 end cell row 1 cell space space space 1 end cell end table close vertical bar space space space space space space equals 2 space left parenthesis negative 1 plus 1 right parenthesis space plus space 1 left parenthesis negative 3 plus 1 right parenthesis space minus space 1 left parenthesis 3 minus 1 right parenthesis space equals space 0 minus 2 minus 2 space equals space minus 4 not equal to 0 therefore space space 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 1 cell space space space minus 1 end cell row cell space space 1 end cell cell space space space space minus 1 end cell end table close vertical bar comma space space space space minus space open vertical bar table row 3 cell space space space minus 1 end cell row 1 cell space space minus 1 end cell end table close vertical bar comma space space space space space minus open vertical bar table row 3 cell space space space space space space 1 end cell row 1 cell space space space space space space 1 end cell end table close vertical bar
i.e. 0,  2,  2 respectively
Co-factors of the elements of second row of | A | are
negative open vertical bar table row cell negative 1 end cell cell space space space minus 1 end cell row 1 cell space space space minus 1 end cell end table close vertical bar comma space space space space open vertical bar table row 2 cell space space space minus 1 end cell row 1 cell space space space minus 1 end cell end table close vertical bar comma space space space minus open vertical bar table row 2 cell space space space minus 1 end cell row 1 cell space space space space space 1 end cell end table close vertical bar
i.e. – 2, – 1, – 3 respectively
Co-factors of the elements of third row of | A | are
open vertical bar table row cell negative 1 end cell cell space space space minus 1 end cell row 1 cell space space space space minus 1 end cell end table close vertical bar comma space space space minus open vertical bar table row 2 cell space space space minus 1 end cell row 3 cell space space minus 1 end cell end table close vertical bar comma space space open vertical bar table row 2 cell space space space minus 1 end cell row 3 cell space space space space space space 1 end cell end table close vertical bar
i.e.  2, 1, 5 respectively
space therefore space space space adj. space straight A space equals space open square brackets table row 0 cell space space space space space 2 end cell cell space space space space space space 2 end cell row cell negative 2 end cell cell space space minus 1 end cell cell space space minus 3 end cell row 2 cell space space minus 1 end cell cell space space space space 5 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 0 cell space space space minus 2 end cell cell space space 2 end cell row 2 cell space space minus 1 end cell cell space space 1 end cell row 2 cell space minus 3 end cell cell space 5 end cell end table close square brackets space therefore 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 negative 1 fourth open square brackets table row 0 cell space space space minus 2 end cell cell space space space 2 end cell row 2 cell space space space minus 1 end cell cell space space space space 1 end cell row 2 cell space minus 3 end cell cell space space space space 5 end cell end table close square brackets Now comma space space AX space equals space straight B rightwards double arrow space space 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 open square brackets table row straight x row straight y row straight z end table close square brackets space equals space minus 1 fourth open square brackets table row cell 0 minus 14 plus 6 end cell row cell 14 minus 7 minus 3 end cell row cell 14 minus 21 plus 15 end cell end table close square brackets space space space space space 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 space open square brackets table row cell space space space space space 2 end cell row cell space minus 1 end cell row cell space space minus 2 end cell end table close square brackets therefore space space space space space space solution space is space straight x space equals space 2 comma space space space space straight y space equals space minus 1 comma space space straight z space equals space minus 2

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