Question

Determine uma matriz X que satisfaça a equação matricial XA=B$C^{t}$ onde $A= \left[\begin{array}{ccc}-1&2\\5&2\end{array}\right] B= \left[\begin{array}{ccc}1&0&-3\end{array}\right] C= \left[\begin{array}{ccc}2&0&1\\-4&0&2\end{array}\right]$

Answer 1

Explicação passo-a-passo:

Temos que:

• $\sf C=\left[\begin{array}{ccc} \sf 2& \sf 0& \sf 1 \\ \sf -4& \sf 0& \sf 2\end{array}\right]~\rightarrow~C^t=\left[\begin{array}{cc} \sf 2& \sf -4 \\ \sf 0& \sf 0 \\ \sf 1& \sf 2\end{array}\right]$

• $\sf B\cdot C^t=\left[\begin{array}{ccc}\sf 1& \sf 0&\sf -3\end{array}\right]\cdot\left[\begin{array}{cc} \sf 2& \sf -4 \\ \sf 0& \sf 0 \\ \sf 1& \sf 2\end{array}\right]$

$\sf B\cdot C^t=\left[\begin{array}{cc} \sf 1\cdot2+0\cdot0+(-3)\cdot1 & \sf 1\cdot(-4)+0\cdot0+(-3)\cdot2 \end{array}\right]$

$\sf B\cdot C^t=\left[\begin{array}{cc} \sf 2+0-3 & \sf -4+0-6 \end{array}\right]$

$\sf B\cdot C^t=\left[\begin{array}{cc} \sf -1 & \sf -10 \end{array}\right]$

• Seja $\sf X=\left[\begin{array}{cc} \sf a& \sf b \end{array}\right]$

$\sf X\cdot A=\left[\begin{array}{cc} \sf a& \sf b \end{array}\right]\cdot\left[\begin{array}{ccc}-1&2\\5&2\end{array}\right]$

$\sf X\cdot A=\left[\begin{array}{cc} \sf -a+5b& \sf 2a+2b \end{array}\right]$

Assim:

$\sf \left[\begin{array}{cc} \sf -a+5b& \sf 2a+2b \end{array}\right]=\left[\begin{array}{cc} \sf -1 & \sf -10 \end{array}\right]$

Devemos ter:

$\sf \begin{cases} \sf -a+5b=-1 \\ \sf 2a+2b=-10 \end{cases}$

Da primeira equação:

$\sf -a+5b=-1$

$\sf a=5b+1$

Substituindo na segunda equação:

$\sf 2\cdot(5b+1)+2b=-10$

$\sf 10b+2+2b=-10$

$\sf 12b=-10-2$

$\sf 12b=-12$

$\sf b=\dfrac{-12}{12}$

$\sf b=-1$

Assim:

$\sf a=5b+1$

$\sf a=5\cdot(-1)+1$

$\sf a=-5+1$

$\sf a=-4$

Logo, $\sf X=\left[\begin{array}{cc} \sf -4& \sf -1 \end{array}\right]$