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Determinants

Question
CBSEENMA12034822
Wired Faculty App

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

Solution

The given equations are
5x – y + z = 4
3x + 2y – 5z = 2
x + 3y – 2z = 5
These equations can be written as
                open square brackets table row 5 cell space space space minus 1 end cell cell space space space 1 end cell row 3 cell space space space space space space 2 end cell cell space minus 5 end cell row 1 cell space space space space space 3 end cell cell space minus 2 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 4 row 2 row 5 end table close square brackets
or   AX = B  where A = straight A space equals space open square brackets table row 5 cell space space minus 1 end cell cell space space space space space space 1 end cell row 3 cell space space space space space 2 end cell cell space space space minus 5 end cell row 1 cell space space space space 3 end cell cell space space minus 2 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 4 row 2 row 5 end table close square brackets
               open vertical bar straight A close vertical bar space equals space open vertical bar table row 5 cell space space minus 1 end cell cell space space space space space space 1 end cell row 3 cell space space space space 2 end cell cell space space minus 5 end cell row 1 cell space space space space 3 end cell cell negative 2 end cell end table close vertical bar space equals space 5 left parenthesis negative 4 plus 15 right parenthesis space minus left parenthesis negative 1 right parenthesis space left parenthesis negative 6 plus 5 right parenthesis space plus space left parenthesis 9 minus 2 right parenthesis space space space space space space space equals 5 left parenthesis 11 right parenthesis plus 1 left parenthesis negative 1 right parenthesis plus 1 left parenthesis 7 right parenthesis space equals space 55 minus 1 plus 7 space equals space 61 space not equal to 0 space space space space space space space rightwards double arrow 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 2 cell space space minus 5 end cell row 3 cell space space minus 2 end cell end table close vertical bar comma space space space space minus open vertical bar table row 3 cell space space space minus 5 end cell row 1 cell space space minus 2 end cell end table close vertical bar comma space space space open vertical bar table row 3 cell space space space 2 end cell row 1 cell space space space 3 end cell end table close vertical bar
 or  11, 1, 7 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 space space 1 end cell row 3 cell space minus 2 end cell end table close vertical bar comma space space space space open vertical bar table row 5 cell space space space space space space space space space 1 end cell row 1 cell space space space space space minus 2 end cell end table close vertical bar comma space space space space minus open vertical bar table row 5 cell space space space minus 1 end cell row 1 cell space space space space 3 end cell end table close vertical bar or space space 1 comma space – space 11 comma space – space 16 space 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 space space space space 1 end cell row cell space 2 end cell cell space space minus 5 end cell end table close vertical bar comma space space space space minus open vertical bar table row 5 cell space space space space 1 end cell row 3 cell space space minus 5 end cell end table close vertical bar comma space space open vertical bar table row 5 cell space space minus 1 end cell row 3 cell space space space space space space 2 end cell end table close vertical bar
or   3, 28, 13 respectively.
therefore space space space space adj. space straight A space equals space open square brackets table row 11 cell space space 1 end cell cell space space 7 end cell row 1 cell negative 11 end cell cell negative 16 end cell row 3 28 cell space 13 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 11 cell space space space space 1 end cell cell space space 3 end cell row 1 cell negative 11 end cell cell space space 28 end cell row 7 cell negative 16 end cell cell space 13 end cell end table close square brackets space 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 61 open square brackets table row 11 cell space space space 1 end cell cell space space 3 end cell row 1 cell space minus 11 end cell cell space space 28 end cell row 7 cell negative 16 end cell cell space space 13 end cell end table close square brackets
Now,   
       AX space equals space straight B 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 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 61 open square brackets table row 11 cell space space space 1 end cell cell space space 3 end cell row 1 cell negative 11 end cell cell space space 28 end cell row 7 cell negative 16 end cell cell space space 13 end cell end table close square brackets space space open square brackets table row 4 row 2 row 5 end table close square brackets 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 1 over 61 open square brackets table row cell 44 plus 2 plus 15 end cell row cell 4 minus 22 plus 140 end cell row cell 28 minus 38 plus 65 end cell end table close square brackets 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 1 row 2 row 1 end table close square brackets rightwards double arrow space space space space space space space space straight x space equals space 1 comma space space space space space straight y space space equals space 2 comma space space space space straight z space equals space 1

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