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Determinants

Question
CBSEENMA12034751
Wired Faculty App

Using matrices, solve the following system of linear equations:
x + 2y – 3z = – 4
2x + 3y + 2z = 2
3x – 3y – 4z = 11 

Solution

The given equations are
x + 2y – 3z = – 4
2x + 3y + 2z = 2
3x – 3y – 4z = 11
These equations can be written as
                    open square brackets table row 1 cell space space space space 2 end cell cell space space minus 3 end cell row 2 cell space space space 3 end cell cell space space space space space space 2 end cell row 3 cell space minus 3 end cell cell space space minus 4 end cell end table close square brackets 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 negative 4 end cell row cell space space space 2 end cell row cell space space 11 end cell end table close square brackets
or space space AX space space equals space straight B space where space straight A space equals space open square brackets table row 1 cell space space space 2 end cell cell space space minus 3 end cell row 2 cell space space space space 3 end cell cell space space space space space space 2 end cell row 3 cell space minus 3 end cell cell space space minus 4 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 straight B space equals space open square brackets table row cell negative 4 end cell row cell space space 2 end cell row cell space space 11 end cell end table close square brackets open vertical bar straight A close vertical bar space equals space open square brackets table row 1 cell space space space space 2 end cell cell space space minus 3 end cell row 2 cell space space space space 3 end cell cell space space space space space 2 end cell row 3 cell negative 3 end cell cell space minus 4 end cell end table close square brackets space equals space 1 space open vertical bar table row cell space space space 3 end cell cell space space space space space space space space 2 end cell row cell negative 3 end cell cell space space space minus 4 end cell end table close vertical bar minus space 2 space open vertical bar table row 2 cell space space space space space space space 2 end cell row 3 cell space space minus 4 end cell end table close vertical bar space plus space left parenthesis negative 3 right parenthesis space open vertical bar table row 2 cell space space space space space space space 3 end cell row 3 cell space space minus 3 end cell end table close vertical bar space space space space space space equals 1 space left parenthesis negative 12 plus 6 right parenthesis space minus space 2 left parenthesis negative 8 minus 6 right parenthesis space minus space 3 left parenthesis negative 6 minus 9 right parenthesis space equals space 1 left parenthesis negative 6 right parenthesis space minus space 2 left parenthesis negative 14 right parenthesis minus 3 left parenthesis negative 15 right parenthesis space space space space space space space equals negative 6 plus 28 plus 45 space equals space 67 space not equal to space 0   
Co-factors of the elements of first row of | A | are
   negative open vertical bar table row cell space space space 2 end cell cell space space space minus 3 end cell row cell negative 3 end cell cell space space space minus 4 end cell end table close vertical bar comma space space space space open vertical bar table row 1 cell space space space minus 3 end cell row 3 cell space space space minus 4 end cell end table close vertical bar comma space space minus open vertical bar table row 1 cell space space space space space 2 end cell row 3 cell space minus 3 end cell end table close vertical bar space space straight i. straight e. space minus left parenthesis negative 8 minus 9 right parenthesis comma space space minus 4 plus 9 comma space space minus left parenthesis negative 3 minus 6 right parenthesis straight i. straight e. space space space space 17 comma space space 5 comma space space 9 space respectively.
Co-factors of the elements of thitd row of | A | are
open vertical bar table row 2 cell space space minus 3 end cell row 3 cell space space space space space 2 end cell end table close vertical bar comma space space space space space minus open vertical bar table row 1 cell space space space minus 3 end cell row 2 cell space space space space space 2 end cell end table close vertical bar comma space space space space open vertical bar table row 1 cell space space space 2 end cell row 2 cell space space 3 end cell end table close vertical bar space space straight i. straight e. space space 4 plus 9 comma space space space space minus left parenthesis 2 plus 6 right parenthesis comma space space 3 minus 4
i.e. 13, – 8, – 1 respectively.
therefore space space space space space space space space space space space space space space adj. space straight A space equals space open square brackets table row cell negative 6 end cell cell space space 14 end cell cell space space minus 15 end cell row 17 cell space space space space 5 end cell cell space space space space space 9 end cell row 13 cell space minus 8 end cell cell space space space space 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 6 end cell cell space space space 17 end cell cell space space space space space 13 end cell row cell space 14 end cell cell space space space 5 end cell cell space space space minus 8 end cell row cell negative 15 end cell cell space space 9 end cell cell space space space minus 1 end cell end table close square brackets therefore space space space space space space space straight A to the power of negative 1 end exponent space 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 67 open square brackets table row cell negative 6 end cell cell space space 17 end cell cell space space 13 end cell row cell space space 14 end cell cell space space 5 end cell cell negative 8 end cell row cell negative 15 end cell cell space 9 end cell cell negative 1 end cell end table close square brackets Now comma space AX space equals space straight B space space space space space rightwards double arrow space space space space space straight X equals straight A to the power of negative 1 end exponent straight B
therefore space space space space space 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 67 open square brackets table row cell negative 6 end cell cell space space 17 end cell cell space space 13 end cell row 14 cell space 5 end cell cell negative 8 end cell row cell negative 15 end cell cell space 9 end cell cell negative 1 end cell end table close square brackets space open square brackets table row cell negative 4 end cell row cell space space 2 end cell row cell space 11 end cell end table close square brackets space rightwards double arrow space space 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 67 open square brackets table row cell 24 plus 34 plus 143 end cell row cell negative 56 plus 10 minus 38 end cell row cell 60 plus 18 minus 11 end cell end table close square brackets 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 1 over 67 open square brackets table row cell space space space space 201 end cell row cell negative 134 end cell row cell space space space space 67 end cell end table close square brackets therefore space space space space 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 open square brackets table row cell space space space space 3 end cell row cell negative 2 end cell row cell space space space 1 end cell end table close square brackets space space space therefore space space space space space space space space space space space straight x space equals space 3 comma space space space space space straight y space equals space minus 2 comma space space space straight z space equals space 1.

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