Question

Puteti sa ma ajutati urgent la asta,va rog? Determinati nr reale x si y,stiind ca A+2B=I2,unde I2=(1 0 0 1)
Matricea A=( 1 2 4 1) si B( 0 x y 0)

Answer 1

$\displaystyle A+2B=I_2 \\ \\ I_2= \left(\begin{array}{ccc}1&0\\0&1\\\end{array}\right) \\ \\ A= \left(\begin{array}{ccc}1&2\\4&1\\\end{array}\right)~~~~~~~~~~~~~~~~~B= \left(\begin{array}{ccc}0&x\\y&0\\\end{array}\right)$
$\displaystyle A+2B= \left(\begin{array}{ccc}1&2\\4&1\\\end{array}\right) +2 \cdot \left(\begin{array}{ccc}0&x\\y&0\\\end{array}\right) =\left(\begin{array}{ccc}1&2\\4&1\\\end{array}\right) +\left(\begin{array}{ccc}2 \cdot 0&2 \cdot x\\2 \cdot y&2 \cdot 0\\\end{array}\right) = \\ \\ =\left(\begin{array}{ccc}1&2\\4&1\\\end{array}\right) +\left(\begin{array}{ccc}0&2x\\2y&0\\\end{array}\right) =\left(\begin{array}{ccc}1&2+2x\\4+2y&1\\\end{array}\right)$
$\displaystyle \left(\begin{array}{ccc}1&2+2x\\4+2y&1\\\end{array}\right) =\left(\begin{array}{ccc}1&0\\0&1\\\end{array}\right) \\ \\ 2+2x=0 \Rightarrow 2x=0-2 \Rightarrow 2x=-2 \Rightarrow x=- \frac{2}{2} \Rightarrow \boxed{x=-1} \\ \\ 4+2y=0 \Rightarrow 2y=0-4 \Rightarrow 2y=-4 \Rightarrow y=- \frac{4}{2} \Rightarrow \boxed{y=-2}$