Question

Considere uma função matricial f cujo domínio é o conjunto das matrizes quadradas

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Answer 1

A soma dos elementos da matriz dada por $f\left(\left[\begin{array}{cc}1&2\\1&0\end{array}\right]\right)$ é C) 6.

Para encontrar a respostas, devemos calcular a matriz que resulta do cálculo de $f\left(\left[\begin{array}{cc}1&2\\1&0\end{array}\right]\right)$.

Para isso, temos que:

$f(X) = X^2 + 2I\cdot X - C\\I = \left[\begin{array}{cc}1&0\\0&1\end{array}\right] \\C = \left[\begin{array}{cc}1&3\\2&4\end{array}\right]$

Calculando X²:

$X^2 = \left[\begin{array}{cc}1&2\\1&0\end{array}\right]\cdot \left[\begin{array}{cc}1&2\\1&0\end{array}\right] = \left[\begin{array}{cc}1\cdot 1+2\cdot 1&1\cdot 2+2\cdot 0\\1\cdot 1+0\cdot 1&1\cdot 2+0\cdot 0\end{array}\right]= \left[\begin{array}{cc}3&2\\1&2\end{array}\right]$

Calculando 2I·X:

$2\cdot I\cdot X = 2\cdot \left[\begin{array}{cc}1&0\\0&1\end{array}\right]\cdot \left[\begin{array}{cc}1&2\\1&0\end{array}\right] = 2\cdot\left[\begin{array}{cc}1\cdot 1+0\cdot 1&1\cdot 2+0\cdot 0\\0\cdot 1+1\cdot 1&0\cdot 2+1\cdot 0\end{array}\right]= \left[\begin{array}{cc}2&4\\2&0\end{array}\right]$

Calculando f(X), temos:

$f\left(\left[\begin{array}{cc}1&2\\1&0\end{array}\right]\right)=\left[\begin{array}{cc}3&2\\1&2\end{array}\right]+\left[\begin{array}{cc}2&4\\2&0\end{array}\right]-\left[\begin{array}{cc}1&3\\2&4\end{array}\right]=\left[\begin{array}{cc}4&3\\1&-2\end{array}\right]$

Somando os elementos, encontramos:

4 + 3 + 1 - 2 = 6

Resposta: C