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Determinants

Question
CBSEENMA12034942
Wired Faculty App

Investigate for what values of a and b the simultaneous equations:
x + y + z = 6
x + 2y + 3z = 10
x + 2y + az = b have a unique solution.

Solution

The given equations are
x + y + z = 6
x + 2y + 3z = 10
x + 2y + az = b
These equations can be written as
                               open square brackets table row 1 cell space space space 1 end cell cell space space space 1 end cell row 1 cell space space space 2 end cell cell space space space 3 end cell row 1 cell space space space 2 end cell cell space space space straight a end cell end table close square brackets 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 16 row 10 row straight b 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 1 cell space space space 1 end cell cell space space 1 end cell row 1 cell space space 2 end cell cell space space 3 end cell row 1 cell space 2 end cell cell space space straight a 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 space equals space open square brackets table row 6 row 10 row straight b 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 space 1 end cell row 1 cell space space space 2 end cell cell space space 3 end cell row 1 cell space space space 2 end cell cell space straight a space end cell end table close vertical bar space equals space 1 left parenthesis 2 straight a minus 6 right parenthesis space minus space 1 space left parenthesis straight a minus 3 right parenthesis space plus space 1 space left parenthesis 2 minus 2 right parenthesis space space space space space space equals space space 2 straight a minus 6 space minus straight a plus 3 plus 0 space equals space straight a minus 3
Given system of equation has a unique solution
when | A | ≠ 0 i.e., a – 3 ≠ i.e., a ≠ 3, and b and may have any value.

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