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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

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257.
Solve the following equation by matrix method:
$2 straight x plus straight y plus straight z space equals space 1 straight x minus 2 straight y minus straight z space equals 3 over 2 space space 3 straight y minus 5 straight z space space equals 9$

The given equations are
$2 straight x plus straight y plus straight z space equals space 1 straight x minus 2 straight y minus straight z space equals space 3 over 2 3 straight y minus 5 straight z space equals space 9$
These equations can be written as
$open square brackets table row 2 cell space space space space space space space 1 end cell cell space space space space space space 1 end cell row 2 cell space space space minus 4 end cell cell space space space minus 2 end cell row 0 cell space space space space space 3 end cell cell space space minus 5 end cell end table close square brackets space space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row 1 row 3 row 9 end table close square brackets$

$or space space space space space 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 1 end cell cell space space space space space space 1 end cell row 2 cell space space minus 4 end cell cell space space minus 2 end cell row 0 cell space space space space space space 3 end cell cell space space space minus 5 end cell end table close square brackets comma space space space straight X space equals space space open square brackets table row straight x row straight y row straight z end table close square brackets comma space space straight B space equals space open square brackets table row 1 row 3 row 9 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 1 end cell cell space space space space space space 1 end cell row 2 cell space space minus 4 end cell cell space space space minus 2 end cell row 0 cell space space space space space 3 end cell cell space space minus 5 end cell end table close vertical bar space equals space 2 open vertical bar table row cell negative 4 end cell cell space space space minus 2 end cell row 3 cell space space minus 5 end cell end table close vertical bar minus 1 open vertical bar table row 2 cell space space minus 2 end cell row 0 cell space space space minus 5 end cell end table close vertical bar space plus space 1 space open vertical bar table row 2 cell space space minus 4 end cell row 0 cell space space space space 3 end cell end table close vertical bar space space space space space space space equals 2 left parenthesis 20 plus 6 right parenthesis space minus space 1 left parenthesis negative 10 minus 0 right parenthesis space plus space 1 left parenthesis 6 plus 0 right parenthesis space equals space 52 plus 10 plus 6 space equals space space 68 space not equal to 0 therefore space space space 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 straight A subscript 11 space equals space open vertical bar table row cell negative 4 end cell cell space space space minus 2 end cell row 3 cell space minus 5 end cell end table close vertical bar space equals space 20 plus 6 space equals space 26 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 minus 2 end cell row 0 cell space space space minus 5 end cell end table close vertical bar space equals space minus left parenthesis negative 10 plus 0 right parenthesis space equals space 10 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 4 end cell row 0 cell space space space space space space 3 end cell end table close vertical bar space equals space 6 plus 0 space equals space 6 space space space space space space space space space straight A subscript 21 space equals space minus open vertical bar table row 1 cell space space space space 1 end cell row 3 cell space space space minus 5 end cell end table close vertical bar space equals space minus left parenthesis negative 5 minus 3 right parenthesis space equals space 8 space space space space space space space space space straight A subscript 22 space equals space space open vertical bar table row 2 cell space space space space space space 1 end cell row 0 cell space space space minus 5 end cell end table close vertical bar space equals space minus 10 minus 0 space equals space minus 10 space space space space space space space space space space space space space space space space space space$
$straight A subscript 23 space equals negative open vertical bar table row 2 cell space space space 1 end cell row 0 cell space space space 3 end cell end table close vertical bar space equals space minus left parenthesis 6 minus 0 right parenthesis space equals space minus 6$

$straight A subscript 31 space equals space open vertical bar table row cell space space 1 end cell cell space space space space space space 1 end cell row cell negative 4 end cell cell space space minus 2 end cell end table close vertical bar space equals space minus 2 plus 4 space equals space 2 straight A subscript 32 space equals space minus open vertical bar table row 2 cell space space space space space space 1 end cell row 2 cell space space space minus 2 end cell end table close vertical bar space equals space minus left parenthesis negative 4 minus 2 right parenthesis space equals space 6 straight A subscript 33 space equals space open vertical bar table row 2 cell space space space space space space space 1 end cell row 2 cell space space space minus 4 end cell end table close vertical bar space equals space minus 8 minus 2 space equals space minus 10 space space space adj. space straight A space equals space open square brackets table row 26 cell space space space space space 10 end cell cell space space space space space 6 end cell row 8 cell space minus 10 end cell cell space space minus 6 end cell row 2 cell space space 6 end cell cell space space minus 10 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 26 cell space space space 8 end cell cell space space space 2 space end cell row 10 cell space minus 10 end cell cell space space 6 end cell row 6 cell negative 6 end cell cell negative 10 end cell end table close square brackets 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 equals space space 1 over 68 open square brackets table row 26 cell space space space space space space 8 end cell cell space space space space 2 end cell row 10 cell space space minus 10 end cell cell space space 6 end cell row 6 cell space space minus 6 end cell cell space minus 10 end cell end table close square brackets Now space AX space equals space straight B 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 space 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 1 over 68 open square brackets table row 26 cell space space space space 8 end cell cell space space space 2 end cell row 0 cell space minus 10 end cell cell space space space space 6 end cell row 6 cell negative 6 end cell cell negative 10 end cell end table close square brackets space space space space space open square brackets table row 1 row 3 row 9 end table close square brackets rightwards double arrow space space 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 1 over 68 open square brackets table row cell 26 plus 24 plus 18 end cell row cell 10 minus 30 plus 54 end cell row cell 6 minus 18 minus 90 end cell end table close square brackets space space space space rightwards double arrow 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 1 over 68 open square brackets table row cell space space 68 end cell row cell space 34 end cell row cell negative 102 end cell end table close square brackets rightwards double arrow space space space 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 space equals space open square brackets table row 1 row cell 1 half end cell row cell negative 3 over 2 end cell end table close square brackets therefore space space space space space straight x space equals space 1 comma space space space space space space straight y space equals space 1 half comma space space space straight z space equals space minus 3 over 2.$