Question

Seja A=$\left[\begin{array}{cc}1&2\\0&-1\\\end{array}\right]$ e B=$\left[\begin{array}{cc}-1&3\\2&-6\\\end{array}\right]$. Seja X uma matriz tal que $(XA)^{T}=B.$ . Determine X.

Answer 1

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

$XA= \left[\begin{array}{cc}1a+0b&2a-b\\c+0d&2c-d\end{array}\right] \\ \\ XA= \left[\begin{array}{cc}a&2a-b\\c&2c-d\end{array}\right] \\ \\ (XA)^t= \left[\begin{array}{cc}a&c\\2a-b&2c-d\end{array}\right]$

Como $(XA)^t=B$, os elementos correspondentes das duas matrizes são iguais.

$a=-1 \\ \\ c = 3 \\ \\ 2a-b = 2 \\ -2-b=2 \\ b=-4 \\ \\ 2c-d=-6 \\ 6-d=-6 \\ d=12 \\ \\ X= \left[\begin{array}{cc}-1&-4\\3&12\end{array}\right]$