Question

Determinantes

Determine x tal que:

a) |3x x+2|=11
    |5     4 |

b) |2x-1 x+3|=0
    | 4      x  |

c) |x  3  1|
    |1  4  5|=55
    |x - 2 3|

d) |  3   1  4|
    |x+2 -1 5| =35
    |  2   3 -2|

Se possivel deixar os calculos eu agradeço.

Answer 1

E aí Ganguiamente,

para equação matricial 2x2, use o mesmo procedimento, usado para de terminante de 2ª ordem:

$\left|\begin{array}{ccc}3x&x+2\\5&4\\\end{array}\right|=11\\\\\\ 4*3x-5(x+2)=11\\ 12x-5x-10=11\\ 7x=11+10\\ 7x=21\\ x=21/7\\ x=3\\\\ \boxed{S=\{7\}}$

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$\left|\begin{array}{ccc}2x-1&x+3\\4&x\\\end{array}\right|=0\\\\\\ (2x-1)*x-4(x+3)=0\\ 2 x^{2} -x-4x-12=0\\ 2 x^{2} -5x-12=0$

$\boxed{\Delta=b ^{2}-4ac}\\\\ \Delta=(-5)^{2}-4*2*(-12)\\ \Delta=25+96\\ \Delta=121$

$\boxed{x= \frac{-b\pm \sqrt{\Delta} }{2a}}\\\\\\ x= \frac{-(-5)\pm \sqrt{121} }{2*2}\to~x= \frac{5\pm11}{4}\to\begin{cases}x'= \frac{5-11}{4}\to~x'= \frac{-6}{4}\to~x'=- \frac{3}{2}\\\\ x''= \frac{5+11}{4}\to~x''= \frac{16}{4}\to~x''=4 \end{cases}$

Portanto, a solução da equação matricial será:

$\boxed{S=\{- \frac{3}{2} ,~4\}}$

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E para matricial 3x3, utilize o mesmo procedimento usado para matriz de 3ª ordem, (regra de Sarruz):

$\left|\begin{array}{ccc}x&3&1\\1&4&5\\x&-2&3\end{array}\right| \left\begin{array}{ccc}x&3\\1&4\\x&-2\end{array}\right=55\\\\\\ d.p.~\to~12x+15x-2\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=55\\ d.s.~\to~-4x+10x-9\\\\ (d.p.)+(d.s.)~\to~33x-11=55\\\\ 33x=55+11\\ 33x=66\\ x=66/33\\ x=2\\\\ \boxed{S=\{2\}}$

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$\left|\begin{array}{ccc}3&1&4\\x+2&-1&5\\2&3&-2\end{array}\right| \left\begin{array}{ccc}3&1\\x+2&-1\\2&3\end{array}\right=35\\\\\\ d.p.\to~6+10+12x+24\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=35\\ d.s.\to~8-45+2x+4\\\\\\ d.p.\to~40+12x\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~=35\\ d.s.\to~-33+2x\\\\\\ (d.p.)+(d.s.)~\to~7+14x=35\\ 14x=35-7\\ 14x=28\\ x=28/14\\ x=2\\\\\\ \boxed{S=\{2\}}$


Espero ter ajudado e tenha ótimos estudos =))