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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

93 Views

243.

Use matrix method to solve the system of equations:
x + 2y = 4
2x + 5y = 9

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244.
Use matrix method to solve the system of equations:
2x + 3y = 5
3x - y = 2

The given equations are
2x + 3y = 5
3x – y = 2
These equations can be written as
   $open square brackets table row 2 cell space space space space space space 3 end cell row 3 cell space space minus 1 end cell end table close square brackets space open square brackets table row straight x row straight y end table close square brackets space equals space open square brackets table row 5 row 2 end table close square brackets$
or $AX space equals space straight B space where space straight A space equals space open square brackets table row 2 cell space space space space space 3 end cell row 3 cell space minus 1 end cell end table close square brackets comma space space straight X space equals space open square brackets table row straight x row straight y end table close square brackets comma space space straight B space equals space open square brackets table row 5 row 2 end table close square brackets$
   $open vertical bar straight A close vertical bar space equals space open vertical bar table row 2 cell space space space space space space 3 end cell row 3 cell space space minus 1 end cell end table close vertical bar comma space space space equals negative 2 minus 9 space equals space minus 11 space not equal to space 0$
$therefore space space space straight A to the power of negative 1 end exponent space exists.$
   $straight A subscript 11 space equals space minus 1 comma space space space space space straight A subscript 12 space equals negative 3 comma space space space straight A subscript 21 space equals space minus 3 comma space space space straight A subscript 22 space equals space 2$
$therefore space space space space space space space space space adj. space straight A space equals space open square brackets table row cell negative 1 end cell cell space space space minus 3 end cell row cell negative 3 end cell cell space space space space space space 2 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row cell negative 1 end cell cell space space space minus 3 end cell row cell negative 3 end cell cell space space space space space 2 end cell end table close square brackets space space space space space 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 negative 1 over 11 open square brackets table row cell negative 1 end cell cell space space space minus 3 end cell row cell negative 3 end cell cell space space space space space 2 end cell end table close square brackets Now space AX space equals space straight B 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 rightwards double arrow space space space space space space space open square brackets table row straight x row straight y end table close square brackets space equals negative 1 over 11 open square brackets table row cell negative 1 end cell cell space space space minus 3 end cell row cell negative 3 end cell cell space space space space space space 2 end cell end table close square brackets space open square brackets table row 5 row 2 end table close square brackets space space rightwards double arrow space space space open square brackets table row straight x row straight y end table close square brackets space equals space minus 1 over 11 open square brackets table row cell negative 5 minus 6 end cell row cell negative 15 plus 4 end cell end table close square brackets space space space rightwards double arrow space space space space space space space space open square brackets table row straight x row straight y end table close square brackets space equals space minus 1 over 11 open square brackets table row cell negative 11 end cell row cell negative 11 end cell end table close square brackets space space space space space 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 space open square brackets table row 1 row 1 end table close square brackets therefore space space space space straight x space equals space 1 comma space space straight y space equals space 1$