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Determinants

Question
CBSEENMA12034810
Wired Faculty App

Solve (Use matrix method):
x + y = 0
y + z = 1
z + x = 3   

Solution

The given equations are
x + y + Oz =0
Ox + y + z = 1
x + 0y + z = 3
These equations can be written as
open square brackets table row 1 cell space space 1 end cell cell space space 0 end cell row 0 cell space space 1 end cell cell space space 1 end cell row 1 cell space space 0 end cell cell 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 space open square brackets table row 0 row 1 row 3 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 1 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space 1 end cell cell space space 1 end cell row 1 cell space space 0 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 thin space space straight B space equals space open square brackets table row 0 row 1 row 3 end table close square brackets
open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space 1 end cell cell space 1 end cell row 1 cell space space 0 end cell cell space 1 end cell end table close vertical bar space equals space 1 open vertical bar table row 1 cell space space 1 end cell row 0 cell space space 1 end cell end table close vertical bar space minus space 1 open vertical bar table row 0 cell space space space 1 end cell row 1 cell space space space 1 end cell end table close vertical bar plus 0 open vertical bar table row 0 cell space space space 1 end cell row 1 cell space space space 0 end cell end table close vertical bar space space space space space space space equals 1 space left parenthesis 1 minus 0 right parenthesis space minus space 1 left parenthesis 0 minus 1 right parenthesis space plus space 0 left parenthesis 0 minus 1 right parenthesis space equals space 1 left parenthesis 1 right parenthesis space minus space 1 left parenthesis negative 1 right parenthesis space plus space 0 left parenthesis negative 1 right parenthesis space space space space space space space equals 1 plus 1 plus 0 space equals space 2 not equal to 0 space space space space therefore 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 1 end cell row 0 cell space space 1 end cell end table close vertical bar comma space space space minus open vertical bar table row 0 cell space space space 1 end cell row 1 cell space space space 1 end cell end table close vertical bar comma space space space open vertical bar table row 0 cell space space space 1 end cell row 1 cell space space space space 0 end cell end table close vertical bar
i.e.   1, 1 1 respectively.
Co-factors of the elements of second row of | A | are
therefore space space space space space space space adj. space straight A space equals space open square brackets table row cell space space 1 end cell cell space space space 1 end cell cell space space space minus 1 end cell row cell negative 1 end cell cell space space space space 1 end cell cell space space space space space space space 1 end cell row cell space space 1 end cell cell space minus 1 end cell cell space space minus 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 1 cell space space minus 1 end cell cell space space space space 1 end cell row 1 cell space space space space 1 end cell cell space minus 1 end cell row cell negative 1 end cell cell space space space 1 end cell cell space space space space space 1 end cell end table close square brackets
i.e. -1, 1, 1 respectively.
Co-factors of the elements of third row of | A | are
open vertical bar table row 1 cell space space 0 end cell row 1 1 end table close vertical bar comma space space space minus open vertical bar table row 1 cell space space space 0 end cell row 0 cell space space 1 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space 1 end cell row 0 cell space space 1 end cell end table close vertical bar
i.e. 1,  -1,  1 respectively.

    or    AX space equals space straight B space space space where space straight A space equals open square brackets table row 1 cell space space space 1 end cell cell space space 0 end cell row 0 cell space space 1 end cell cell space space 1 end cell row 1 cell space space 0 end cell cell space space 1 end cell end table close square brackets comma 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 0 row 1 row 3 end table close square brackets
Co-factors of the elements of first row of | A | are

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