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Determinants

Short Answer Type

241.

Use matrix method to solve the system of equations:
2x + 3y = – 1
x + 2y = 2

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242.
Use matrix method to solve the system of equations:
5x - 7y = 2
7x - 5y = 3

The given equations are
5x – 7y = 2
1x – 5y = 3
The equations can be written as
$open square brackets table row 5 cell space space space minus 7 end cell row 7 cell space space minus 5 end cell end table close square brackets space open square brackets table row straight x row straight y end table close square brackets space equals space open square brackets table row 2 row 3 end table close square brackets$
or AX = B where $straight A space equals space open square brackets table row 5 cell space space space minus 7 end cell row 7 cell space space minus 5 end cell end table close square brackets space space comma space space straight X space equals space open square brackets table row straight x row straight y end table close square brackets comma space space straight B space equals space open square brackets table row 2 row 3 end table close square brackets$
$open vertical bar straight A close vertical bar space equals space open vertical bar table row 5 cell space space space minus 7 end cell row 7 cell space space minus 5 end cell end table close vertical bar space equals space minus 25 plus 49 space equals space 24 space not equal to space 0 space space space rightwards double arrow space space space space straight A to the power of negative 1 end exponent space exists adj. space straight A space equals space open square brackets table row cell negative 5 end cell cell space space space minus 7 end cell row 7 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 cell negative 5 end cell cell space space space space 7 end cell row cell negative 7 end cell cell space space space space 5 end cell end table close square brackets 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 24 open square brackets table row cell negative 5 end cell cell space space space 7 end cell row cell negative 7 end cell cell space space 5 end cell end table close square brackets Now comma space AX space equals space straight B space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow 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 end table close square brackets space equals 1 over 24 open square brackets table row cell negative 5 end cell cell space space space 7 end cell row cell negative 7 end cell cell space space 5 end cell end table close square brackets open square brackets table row 2 row 3 end table close square brackets 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 end table close square brackets space equals space 1 over 24 open square brackets table row cell negative 10 plus 21 end cell row cell negative 14 plus 15 end cell end table close square brackets rightwards double arrow space space space open square brackets table row straight x row straight y end table close square brackets space equals space open square brackets table row cell 11 over 24 end cell row cell 1 over 24 end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space straight x space equals space 11 over 24 comma space space straight y space equals space 1 over 24 space is space the space solution$