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Determinants

Long Answer Type

271.

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

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272.

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

75 Views

273.

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

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274. Use matrix method to solve the following system of equations:
y + 2z = 4
2z + x = 5
x + 2y = 7

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275. Use matrix method to solve the following system of equations:
3x – 4y + 2z = – 1
2x + 3y + 5z = 7
x + z = 2

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

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277. Use matrix method to solve the following system of equations:
x + y + z = 3
x + 2y + 3z = 4
x + 4y + 9z = 6

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

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279. Use matrix method to solve the following system of equations:
x + 2y – 3z = 6
3x + 2y – 2z = 3
2x – y + z = 2

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

These equations can be written as
x + y + z = 5    ...(1)
2x + y – z = 2    ...(2)
2x – y + z = 2    ...(3)
These equations can be written as

open square brackets table row 1 cell space space 1 space space end cell cell space space space 1 end cell row 2 cell space 1 end cell cell negative 1 end cell row 2 cell negative 1 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 5 row 2 row 2 end table close square brackets

or space space space 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 space space 1 end cell row 2 cell space space space space 1 space end cell cell space minus 1 end cell row 2 cell negative 1 end cell cell space space space space 1 end cell end table close square brackets comma space space space straight X 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 5 row 2 row 2 end table close square brackets space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space 1 end cell cell space space space 1 end cell row 2 cell space space 1 end cell cell negative 1 end cell row 2 cell negative 1 end cell cell space space 1 end cell end table close vertical bar space equals space 1 open vertical bar table row 1 cell space space space minus 1 end cell row cell negative 1 end cell cell space space space space 1 end cell end table close vertical bar minus 1 open vertical bar table row 2 cell space space minus 1 end cell row 2 cell space space space space 1 end cell end table close vertical bar plus 1 open vertical bar table row 2 cell space space space space space space 1 end cell row 2 cell space space minus 1 end cell end table close vertical bar space space space space equals space 1 left parenthesis 1 minus 1 right parenthesis minus 1 left parenthesis 2 plus 2 right parenthesis space plus space 1 left parenthesis negative 2 minus 2 right parenthesis space space space space space equals 0 minus 4 minus 4 space equals space minus 8 space not equal to space 0 therefore space 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 minus 1 end cell row cell negative 1 end cell cell space space space space space 1 end cell end table close vertical bar comma space space space minus open vertical bar table row 2 cell space space minus 1 end cell row 2 cell space space space space 1 end cell end table close vertical bar comma space space space open vertical bar table row 2 cell space space space space space 1 end cell row 2 cell space minus 1 end cell end table close vertical bar

i.e. 0, –4, –4 respectively
Co-factors of the elements of second row of | A | are
$negative open vertical bar table row 1 cell space space space 1 end cell row cell negative 1 end cell cell space space 1 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space space 1 end cell row 2 cell space space space 1 end cell end table close vertical bar comma space space minus open vertical bar table row 1 cell space space space space space 1 end cell row 2 cell space space minus 1 end cell end table close vertical bar$
i.e. – 2, – 1, 3 respectively
Co-factors of the elements of third row of | A | are
$open vertical bar table row 1 cell space space space space space 1 end cell row 1 cell space minus 1 end cell end table close vertical bar comma space space space space minus open vertical bar table row 1 cell space space space space space space 1 end cell row 2 cell space space minus 1 end cell end table close vertical bar comma space space space open vertical bar table row 1 cell space space space 1 end cell row 2 cell space space 1 end cell end table close vertical bar$
i.e. 2, 3 – 1 respectively.
$adj. space straight A space equals space open square brackets table row 0 cell space space minus 4 end cell cell space space minus 4 end cell row cell negative 2 end cell cell space space minus 1 end cell cell space space space space space 3 end cell row cell negative 2 end cell cell space space space 3 end cell cell negative 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 0 cell space space space minus 2 end cell cell space space minus 2 end cell row cell negative 4 end cell cell space space minus 1 end cell cell space space space space space 3 end cell row cell negative 4 end cell cell space space space 3 end cell cell space minus 1 end cell end table close square brackets therefore 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 minus 1 over 8 open square brackets table row 0 cell space space space space minus 2 end cell cell space space minus 2 end cell row cell negative 4 end cell cell space space space minus 1 end cell cell space space space space space space 3 end cell row cell negative 4 end cell cell space space space 3 end cell cell space space space minus 1 end cell end table close square brackets Now space space 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 space space space space rightwards double arrow space space space straight X equals space straight A to the power of negative 1 end exponent straight B open square brackets table row straight x row straight y row straight z end table close square brackets space equals space minus 1 over 8 open square brackets table row 0 cell space space minus 2 end cell cell space space space minus 2 end cell row cell negative 4 end cell cell space space minus 1 end cell cell space space space space space 3 end cell row cell negative 4 end cell cell space space space 3 end cell cell space space minus 1 end cell end table close square brackets space open square brackets table row 5 row 2 row 2 end table close square brackets 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 minus 1 over 8 space open square brackets table row cell 0 minus 4 minus 4 end cell row cell negative 20 minus 2 plus 6 end cell row cell negative 20 plus 6 minus 2 end cell end table close square brackets space space rightwards double arrow 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 2 end table close square brackets therefore space space space space space solution space is space straight x space equals space 1 comma space space straight y space equals space 2 comma space space space straight z space equals space 2$