Question

Álgebra Linear, Matrizes
Preciso aprender a resolver essa questão para a prova, galera, me ajudem pfvr!
Seja A=(aij) uma matriz quadrada de ordem 2 tal que aij = i - j. Determine X,Y, Z e T para que se tenha:
| x+y   x+z |
| 3x-1   t+z |  =  A

Answer 1

Olá, Gabriel.

$A=(a_{ij})_{2\times2},\text{tal que }a_{ij}=i-j\Rightarrow\\\\A=\left[\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right]=\left[\begin{array}{cc}1-1&1-2\\2-1&2-2\end{array}\right]=\left[\begin{array}{cc}0&-1\\1&0\end{array}\right]$

$\left[\begin{array}{cc}x+y&x+z\\3x-1&t+z\end{array}\right]=A\Rightarrow\left[\begin{array}{cc}x+y&x+z\\3x-1&t+z\end{array}\right]=\left[\begin{array}{cc}0&-1\\1&0\end{array}\right]\Rightarrow\\\\\\ \begin{cases}x+y=0\\x+z=-1\\3x-1=1\\t+z=0\end{cases}\Rightarrow\\\\3x=1+1=2\Rightarrow \boxed{x=\frac23}\\\\ \frac23+y=0\Rightarrow\boxed{y=-\frac23}\\\\ \frac23+z=-1\Rightarrow z=-\frac23-1\Rightarrow\boxed{z=-\frac53}\\\\ t-\frac53=0\Rightarrow\boxed{t=\frac53}$