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Determinants

Long Answer Type

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

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282. Use matrix method to solve the following system of equations:
2x – y - z = 7
3x + y – z = 7
x + y – z = 3

74 Views

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

73 Views

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

75 Views

285. Use matrix method to solve the following system of equations:
x + 2y + z = 7
x + z = 11
2x – 3y = 1

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286. Use matrix method to solve the following system of equations:
x + 2y – 3z = 4
x + 3 + 2z = 2
3x – 3y + 4z = 11

82 Views

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

73 Views

288. Use matrix method to solve the following system of equations:
3x + 14y + 7z = 14
2x – y + 3z = 4
x + 2y – 3z = 0

76 Views

289. Use matrix method to solve the following system of equations:
5x + 3y + z = 16
2x + y + 3 z = 19
x + 2y+ 4 z = 25

The given equations are
5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25
These equations can be written as
$open square brackets table row 5 cell space space space 3 end cell cell space space space 1 end cell row 2 cell space space space 1 end cell cell space space space 3 end cell row 1 cell space space 2 end cell cell space space space 4 end cell end table close square brackets space open square brackets table row straight x row straight y row straight z end table close square brackets space equals space open square brackets table row 16 row 19 row 25 end table close square brackets$

where $straight A equals space open square brackets table row 5 cell space space 3 end cell cell space space 1 end cell row 2 cell space space 1 end cell cell space space 3 end cell row 1 cell space space 2 end cell cell space space 4 end cell end table close square brackets comma space space space straight X space equals space open square brackets table row straight x row straight y row straight z end table close square brackets comma space space space straight B space equals space open square brackets table row 16 row 19 row 25 end table close square brackets$

$Now comma space open vertical bar straight A close vertical bar space equals space open vertical bar table row 5 cell space space 3 end cell cell space space 1 end cell row 2 cell space space 1 end cell cell space space 3 end cell row 1 cell space space 2 end cell cell space space 4 end cell end table close vertical bar space equals space 5 open vertical bar table row 1 cell space space space 3 end cell row 2 cell space space 4 end cell end table close vertical bar space minus space 3 open vertical bar table row 2 cell space space space 3 end cell row 1 cell space space space space 4 end cell end table close vertical bar space plus space 1 open vertical bar table row 2 cell space space 1 end cell row 1 cell space space 2 end cell end table close vertical bar$

$equals space 5 left parenthesis 4 minus 6 right parenthesis space minus space 3 left parenthesis 8 minus 3 right parenthesis space plus 1 thin space left parenthesis 4 minus 1 right parenthesis space equals space minus 10 minus 15 plus 3 space equals space minus 22$
Co-factors of the elements of first row of | A | are
$open vertical bar table row 1 cell space space space 3 end cell row 2 cell space space 4 end cell end table close vertical bar comma space space minus open vertical bar table row 2 cell space space space 3 end cell row 1 cell space space 4 end cell end table close vertical bar comma space space open vertical bar table row 2 cell space space space 1 end cell row 1 cell space space 2 end cell end table close vertical bar$
i.e., – 2, – 5, 3 respectively.
Co-factors of the elements of 2nd row of | A | are
$negative open vertical bar table row 3 cell space space space 1 end cell row 2 cell space space space 4 end cell end table close vertical bar comma space space space space open vertical bar table row 5 cell space space space 1 end cell row 1 cell space space space 4 end cell end table close vertical bar comma space space space space minus open vertical bar table row 5 cell space space space 3 end cell row 2 cell space space 1 end cell end table close vertical bar$
i.e., – 10, 9, – 7 respectively.
Co-factors of the elements of 3rd row of | A | are
$open vertical bar table row 3 cell space space 1 end cell row 1 cell space space 3 end cell end table close vertical bar comma space space minus open vertical bar table row 5 cell space space space 1 end cell row 2 cell space space space 3 end cell end table close vertical bar comma space space open vertical bar table row 5 cell space space space 3 end cell row 2 cell space space space 1 end cell end table close vertical bar$
i.e., 8, – 13, – 1 respectively.
$therefore space space space space space adj. space straight A space equals space open square brackets table row cell negative 2 end cell cell space space space minus 5 end cell cell space space space 3 end cell row cell negative 10 end cell cell space space space space space space 19 end cell cell space space minus 7 end cell row 8 cell space space minus 13 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 cell negative 2 end cell cell space space minus 10 end cell cell space space space space space space 8 end cell row cell negative 5 end cell cell space space space space space space 19 end cell cell space space minus 13 end cell row 3 cell space space space minus 7 end cell cell negative 1 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 minus 1 over 22 open square brackets table row cell negative 2 end cell cell space space space minus 10 end cell cell space space space space 8 end cell row cell negative 5 end cell cell space space space space 19 end cell cell negative 13 end cell row 3 cell space space minus 7 end cell cell negative 1 end cell end table close square brackets Now comma space space space AX space equals space straight B space space space space space space rightwards double arrow space space space space 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 row straight z end table close square brackets space equals space minus 1 over 22 open square brackets table row cell negative 2 end cell cell space space minus 10 end cell cell space space space space space 8 end cell row cell negative 5 end cell cell space space space space 19 end cell cell negative 13 end cell row 3 cell space minus 7 end cell cell negative 1 end cell end table close square brackets space open square brackets table row 16 row 19 row 25 end table close square brackets 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 minus 1 over 22 open square brackets table row cell negative 32 minus 190 plus 200 end cell row cell negative 80 plus 361 minus 325 end cell row cell 48 minus 133 minus 25 end cell 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 minus 1 over 22 open square brackets table row cell negative 22 end cell row cell negative 44 end cell row cell negative 110 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 open square brackets table row 1 row 2 row 5 end table close square brackets 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 1 comma space space space space space space straight y space equals space 2 comma space space space space straight z space equals space 5$