A soma de todos os elementos da matriz X, de ordem 2, dada por
=
é:
a) 2
b) 4
c) 5
d) 6
e) 9
Soluções para a tarefa
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2
Olá!
![\\ \mathsf{\qquad Seja \ X = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \ ent\~ao \ X^t = \begin{bmatrix} a & c \\ b & d \end{bmatrix}.} \\\\ \mathsf{Com \ efeito,} \\\\ \mathsf{2X + X^t = \begin{bmatrix} 3 & 8 \\ 7 & 0 \end{bmatrix}} \\\\\\ \mathsf{\begin{bmatrix} 2a & 2b \\ 2c & 2d \end{bmatrix} + \begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 0 \end{bmatrix}} \\\\\\ \mathsf{\begin{bmatrix} 3a & 2b + c \\ 2c + b & 3d \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 0 \end{bmatrix}} \\ \mathsf{\qquad Seja \ X = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \ ent\~ao \ X^t = \begin{bmatrix} a & c \\ b & d \end{bmatrix}.} \\\\ \mathsf{Com \ efeito,} \\\\ \mathsf{2X + X^t = \begin{bmatrix} 3 & 8 \\ 7 & 0 \end{bmatrix}} \\\\\\ \mathsf{\begin{bmatrix} 2a & 2b \\ 2c & 2d \end{bmatrix} + \begin{bmatrix} a & c \\ b & d \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 0 \end{bmatrix}} \\\\\\ \mathsf{\begin{bmatrix} 3a & 2b + c \\ 2c + b & 3d \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 0 \end{bmatrix}}](https://tex.z-dn.net/?f=%5C%5C+%5Cmathsf%7B%5Cqquad+Seja+%5C+X+%3D+%5Cbegin%7Bbmatrix%7D+a+%26amp%3B+b+%5C%5C+c+%26amp%3B+d+%5Cend%7Bbmatrix%7D%2C+%5C+ent%5C%7Eao+%5C+X%5Et+%3D+%5Cbegin%7Bbmatrix%7D+a+%26amp%3B+c+%5C%5C+b+%26amp%3B+d+%5Cend%7Bbmatrix%7D.%7D+%5C%5C%5C%5C+%5Cmathsf%7BCom+%5C+efeito%2C%7D+%5C%5C%5C%5C+%5Cmathsf%7B2X+%2B+X%5Et+%3D+%5Cbegin%7Bbmatrix%7D+3+%26amp%3B+8+%5C%5C+7+%26amp%3B+0+%5Cend%7Bbmatrix%7D%7D+%5C%5C%5C%5C%5C%5C+%5Cmathsf%7B%5Cbegin%7Bbmatrix%7D+2a+%26amp%3B+2b+%5C%5C+2c+%26amp%3B+2d+%5Cend%7Bbmatrix%7D+%2B+%5Cbegin%7Bbmatrix%7D+a+%26amp%3B+c+%5C%5C+b+%26amp%3B+d+%5Cend%7Bbmatrix%7D+%3D+%5Cbegin%7Bbmatrix%7D+3+%26amp%3B+8+%5C%5C+7+%26amp%3B+0+%5Cend%7Bbmatrix%7D%7D+%5C%5C%5C%5C%5C%5C+%5Cmathsf%7B%5Cbegin%7Bbmatrix%7D+3a+%26amp%3B+2b+%2B+c+%5C%5C+2c+%2B+b+%26amp%3B+3d+%5Cend%7Bbmatrix%7D+%3D+%5Cbegin%7Bbmatrix%7D+3+%26amp%3B+8+%5C%5C+7+%26amp%3B+0+%5Cend%7Bbmatrix%7D%7D)
![\\ \mathsf{\begin{cases} \mathsf{3a = 3 \Rightarrow \boxed{\mathsf{a = 1}}} \\ \mathsf{2b + c = 8 \qquad \qquad \ eq. \ II} \\ \mathsf{2c + b = 7 \qquad \qquad \ eq. \ III} \\ \mathsf{3d = 0 \Rightarrow \boxed{\mathsf{d = 0}}} \end{cases}} \\\\ \mathsf{Resolvendo \ o \ sistema \ formado \ por \ II \ e \ III, \ tiramos \ que: \ \boxed{\mathsf{b = 3}} \ e \ \boxed{\mathsf{c = 2}}.} \\ \mathsf{\begin{cases} \mathsf{3a = 3 \Rightarrow \boxed{\mathsf{a = 1}}} \\ \mathsf{2b + c = 8 \qquad \qquad \ eq. \ II} \\ \mathsf{2c + b = 7 \qquad \qquad \ eq. \ III} \\ \mathsf{3d = 0 \Rightarrow \boxed{\mathsf{d = 0}}} \end{cases}} \\\\ \mathsf{Resolvendo \ o \ sistema \ formado \ por \ II \ e \ III, \ tiramos \ que: \ \boxed{\mathsf{b = 3}} \ e \ \boxed{\mathsf{c = 2}}.}](https://tex.z-dn.net/?f=%5C%5C+%5Cmathsf%7B%5Cbegin%7Bcases%7D+%5Cmathsf%7B3a+%3D+3+%5CRightarrow+%5Cboxed%7B%5Cmathsf%7Ba+%3D+1%7D%7D%7D+%5C%5C+%5Cmathsf%7B2b+%2B+c+%3D+8+%5Cqquad+%5Cqquad+%5C+eq.+%5C+II%7D+%5C%5C+%5Cmathsf%7B2c+%2B+b+%3D+7+%5Cqquad+%5Cqquad+%5C+eq.+%5C+III%7D+%5C%5C+%5Cmathsf%7B3d+%3D+0+%5CRightarrow+%5Cboxed%7B%5Cmathsf%7Bd+%3D+0%7D%7D%7D+%5Cend%7Bcases%7D%7D+%5C%5C%5C%5C+%5Cmathsf%7BResolvendo+%5C+o+%5C+sistema+%5C+formado+%5C+por+%5C+II+%5C+e+%5C+III%2C+%5C+tiramos+%5C+que%3A+%5C+%5Cboxed%7B%5Cmathsf%7Bb+%3D+3%7D%7D+%5C+e+%5C+%5Cboxed%7B%5Cmathsf%7Bc+%3D+2%7D%7D.%7D)
Por fim, efectuamos a soma dos elementos de
. Segue,
![\\ \mathsf{a + b + c + d = 1 + 3 + 2 + 0} \\ \boxed{\boxed{\mathsf{a + b + c + d = 6}}} \\ \mathsf{a + b + c + d = 1 + 3 + 2 + 0} \\ \boxed{\boxed{\mathsf{a + b + c + d = 6}}}](https://tex.z-dn.net/?f=%5C%5C+%5Cmathsf%7Ba+%2B+b+%2B+c+%2B+d+%3D+1+%2B+3+%2B+2+%2B+0%7D+%5C%5C+%5Cboxed%7B%5Cboxed%7B%5Cmathsf%7Ba+%2B+b+%2B+c+%2B+d+%3D+6%7D%7D%7D)
Por fim, efectuamos a soma dos elementos de
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