Question

Dadas as matrizes A= -1  3   e B= 2  -1 determinem:
                                  2  -8          3   0
a) Det A
b) Det B
c) Det (A+B)
d) Det A + det B
                                  

Answer 1

$det~\left[\begin{array}{cc}a&b\\c&d\end{array}\right]=\left|\begin{array}{cc}a&b\\c&d\end{array}\right|=ad-bc$
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a)

$det~\left[\begin{array}{cc}-1&3\\2&-8\end{array}\right]=(-1)(-8)-3\cdot2\\\\\\det~\left[\begin{array}{cc}-1&3\\2&-8\end{array}\right]=8-6\\\\\\\boxed{\boxed{det~\left[\begin{array}{cc}-1&3\\2&-8\end{array}\right]=2}}$

b)
$det~\left[\begin{array}{cc}2&-1\\3&0\end{array}\right]=2\cdot0-(-1)\cdot3\\\\\\det~\left[\begin{array}{cc}2&-1\\3&0\end{array}\right]=0+3\\\\\\\boxed{\boxed{det~\left[\begin{array}{cc}2&-1\\3&0\end{array}\right]=3}}$

c)

Primeiro, vamos achar A + B:

$A+B=\left[\begin{array}{cc}-1&3\\2&-8\end{array}\right]+\left[\begin{array}{cc}2&-1\\3&0\end{array}\right]\\\\\\A+B=\left[\begin{array}{cc}(-1+2)&(3-1)\\(2+3)&(-8+0)\end{array}\right]\\\\\\A+B=\left[\begin{array}{cc}1&2\\5&-8\end{array}\right]$

Achando o determinante dessa matriz:

$det~\left[\begin{array}{cc}1&2\\5&-8\end{array}\right]=1(-8)-2\cdot5\\\\\\det~\left[\begin{array}{cc}1&2\\5&-8\end{array}\right]=-8-10\\\\\\\boxed{\boxed{det~\left[\begin{array}{cc}1&2\\5&-8\end{array}\right]=-18}}$

d)

$det~A+det~B=2+3\\\\\boxed{\boxed{det~A+det~B=5}}$