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Determinants

Long Answer Type

251.

Using matrices, solve the following system of equations:
2x + y – 3r = 13
x + y – z = 6
2x – y + 4z = – 12

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252.

Using matrices, solve the following system of linear equations.
3x + 4y + 2z = 8
2y –3z = 3
x – 2y + 6z = –2

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253. Using matrices, solve the following system of linear equations:
x – y + z = 1
2x – yz = 2
x – 2y – z

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254.

Using matrices, solve the following system of linear equations:
x – y = 3
2x + 3y + 4z = 17
y + 2 z = 7

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255.

Solve by matrix method:
y + 2z = – 8
x + 2y + 3z = – 14
3x + y + z = – 8

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256.

Solve the following system of equations by matrix method:
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4

89 Views

257.

Solve the following equation by matrix method:
$2 straight x plus straight y plus straight z space equals space 1 straight x minus 2 straight y minus straight z space equals 3 over 2 space space 3 straight y minus 5 straight z space space equals 9$

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258. Solve system of linear equations, using matrix method:
x – y + z = 4
2x + y – 3z = 0
x + y + z = 2

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259.
Solve the following system of equations by matrix method:
x – y + z = 2
2x – y = 0
2y – z = 1

The given equations are
x – y + z = 2
2x – y + 0z = 0
0x + 2y– z = 1
These equations can be written as
$open square brackets table row 1 cell space space space minus 1 end cell cell space space space space 1 end cell row 2 cell space space space minus 1 end cell cell space space space space 0 end cell row 0 cell space space space space space 2 end cell cell negative 1 end cell end table close square brackets space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row 2 row 0 row 1 end table close square brackets$

or $AX space equals space straight B space where space straight A space equals space open square brackets table row 1 cell space space space minus 1 end cell cell space space space 0 end cell row 2 cell space space minus 1 end cell cell space 0 end cell row 0 cell space space space space 0 end cell cell space minus 1 end cell end table close square brackets comma space space space space space straight X space equals open square brackets table row straight x row straight y row straight z end table close square brackets space space space space straight B space equals space open square brackets table row 2 row 0 row 1 end table close square brackets$
Now,
$open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space space minus 1 end cell cell space space space space 0 end cell row 2 cell space space minus 1 end cell cell space space space space 0 end cell row 0 cell space space space space 2 end cell cell space minus 1 end cell end table close vertical bar space equals space 1 open vertical bar table row cell negative 1 end cell cell space space space space space space 0 end cell row 2 cell space space minus 1 end cell end table close vertical bar minus left parenthesis negative 1 right parenthesis space open vertical bar table row 2 cell space space space space space 0 end cell row 0 cell space minus 1 end cell end table close vertical bar plus 1 open vertical bar table row 2 cell space space space minus 1 end cell row 0 cell space space space space space 2 end cell end table close vertical bar space space space space equals 1 left parenthesis 1 minus 0 right parenthesis space plus 1 left parenthesis negative 2 minus 0 right parenthesis plus 1 left parenthesis 4 minus 0 right parenthesis space equals space 1 minus 2 plus 4 space equals 3 space not equal to space 0 therefore space space space space straight A to the power of negative 1 end exponent space exists space space space space space space space space space space straight A subscript 11 space equals space open vertical bar table row cell negative 1 end cell cell space space space 0 end cell row 2 cell negative 1 end cell end table close vertical bar space equals space 1 minus 0 space equals space 1 space space space space space space space space space space space straight A subscript 12 space equals space minus open vertical bar table row 2 cell space space space space space 0 end cell row 0 cell space minus 1 end cell end table close vertical bar space space equals negative left parenthesis negative 2 minus 0 right parenthesis space equals space 2 space space space space space space space space space space space straight A subscript 13 space equals space open vertical bar table row 2 cell space space space minus 1 end cell row 0 cell space space space space space 2 end cell end table close vertical bar space equals space 4 minus 0 space equals space 4 space space space space space space space space space space space space straight A subscript 21 space equals space minus open vertical bar table row cell negative 1 end cell cell space space space space space space space space 1 end cell row 2 cell space space space minus 1 end cell end table close vertical bar space equals space minus left parenthesis 1 minus 2 right parenthesis space equals space 1 space space space space space space space space space space space straight A subscript 22 space equals space open vertical bar table row 1 cell space space space space space space space 0 end cell row 0 cell space space space minus 1 end cell end table close vertical bar space equals space minus 1 minus 0 space equals space minus 1 space space space space space space space space space space space straight A subscript 23 space equals space minus open vertical bar table row 1 cell space space space minus 1 end cell row 0 cell space space space space 2 end cell end table close vertical bar space equals space minus left parenthesis 2 minus 0 right parenthesis space equals space minus 2 space space space space space space space space space space space straight A subscript 31 space equals space open vertical bar table row cell negative 1 end cell cell space space space space 1 end cell row cell negative 1 end cell cell space space space space 0 end cell end table close vertical bar space equals space 0 plus 1 space equals space 1 space space space space space space space space space space straight A subscript 32 space equals space minus open vertical bar table row 1 cell space space space space space 1 space space end cell row 2 cell space space 0 end cell end table close vertical bar space equals space minus left parenthesis 0 minus 2 right parenthesis space equals space 2 space space space space space space space space space space straight A subscript 33 space equals space open vertical bar table row 1 cell space space space minus 1 end cell row 2 cell space space space minus 1 end cell end table close vertical bar space equals space minus 1 plus 2 space equals space 1 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$
$therefore space space space adj. space straight A space equals space open square brackets table row 1 cell space space space space space 2 end cell cell space space space space space space 4 end cell row 1 cell space space minus 1 end cell cell space minus 2 end cell row 1 cell space space space 2 end cell cell space space space space space 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 space space space space 1 end cell cell space space space 1 end cell row 2 cell space space minus 1 end cell cell space space space 2 end cell row 4 cell space space minus 2 end cell cell space space space 1 end cell end table close square brackets 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 space 1 third open square brackets table row 1 cell space space space space space 1 end cell cell space space space 1 end cell row 2 cell space minus 1 end cell cell space space space 2 end cell row 4 cell negative 2 end cell cell space space space 1 end cell end table close square brackets Now comma space space space space AX space equals space straight B space space space space space space space space space space space space 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 rightwards double arrow space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space 1 third open square brackets table row 1 cell space space space space space space 1 end cell cell space space space space 1 end cell row 2 cell space space minus 1 end cell cell space space space space 2 end cell row 4 cell space space minus 2 end cell cell space space space space 1 end cell end table close square brackets space open square brackets table row 2 row 0 row 1 end table close square brackets space space space 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 1 third open square brackets table row cell 2 plus 0 plus 1 end cell row cell 4 plus 0 plus 2 end cell row cell 8 plus 0 plus 1 end cell end table close square brackets 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 1 third open square brackets table row 3 row 6 row 9 end table close square brackets space space space space rightwards double arrow space space 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 open square brackets table row 1 row 2 row 3 end table close square brackets therefore space space space space space space space space space space space space space straight x space equals space 1 comma space space space space space straight y space equals space 2 comma space space space space straight z space equals space 3$