Question

02) Considerando-se as matrizes pode-se afirmar que o determinante de M é igual a a) b) 687 2 - (* ± ² :-D) --- ( 7) **+P-(;;)) C) d) e) 20 02 ) Considerando - se as matrizes pode - se afirmar que o determinante de M é igual a a ) b ) 687 2 - ( * ± ² :-D ) --- ( 7 ) ** + P- ( ;; ) ) C ) d ) e ) 20​

Answer 1

Resposta:

Alternativa b)

Explicação passo a passo:

$\displaystyle N=\left[\begin{array}{cc}a+b&a-c\\b-c&a+b\end{array}\right] \\\\P=\left[\begin{array}{cc}1&-2\\3&1\end{array}\right] \\\\N+P = \left[\begin{array}{cc}0&0\\0&0\end{array}\right]$

$\displaystyle \left[\begin{array}{cc}a+b&a-c\\b-c&a+b\end{array}\right]+\left[\begin{array}{cc}1&-2\\3&1\end{array}\right] = \left[\begin{array}{cc}0&0\\0&0\end{array}\right] \\\\\left[\begin{array}{cc}a+b+1&a-c-2\\b-c+3&a+b+1\end{array}\right] = \left[\begin{array}{cc}0&0\\0&0\end{array}\right]$

a+b+1=0 => a = -b -1 (I)

a-c-2=0 => a +0b-c=2 (II)

b-c+3=0 => 0a+b-c = -3 (III)

a+b+1=0

Substituindo (I) em (II):

-b-1-c=2

b+c= -3 => b = -3-c (IV)

Substituindo (IV) em (III):

-3-c-c=-3

-2c=-3+3

-2c=0

c=0

Substituindo c= 0 em (IV)

b = -3-0

b = -3

Substituindo b= -3 em (I)

a= -(-3)-1

a=3-1

a=2

$M =\left[\begin{array}{ccc}a&b&c\\b&a&b\\a&c&a\end{array}\right] = \left[\begin{array}{ccc}2&-3&0\\-3&2&-3\\2&0&2\end{array}\right]$

$\displaystyle det=\left[\begin{array}{ccc}2&-3&0\\-3&2&-3\\2&0&2\end{array}\right] \\\\\\det=2.2.2+(-3).(-3).2+(-3).0.0-2.2.0-(-3).(-3).2-0.(-3).2=8 + 18 + 0 + 0-18 + 0=8$