Question

Help !! Dau coroana

Answer 1

Răspuns:

Explicație pas cu pas:

$A=\left(\begin{array}{ccc}2&1\\1&2\\\end{array}\right)$

$A^{2} =\left(\begin{array}{ccc}2&1\\1&2\\\end{array}\right)\cdot \left(\begin{array}{ccc}2&1\\1&2\\\end{array}\right)=\left(\begin{array}{ccc}5&4\\4&5\\\end{array}\right)$

$A^{2} -4A=\left(\begin{array}{ccc}5&4\\4&5\\\end{array}\right)-\left(\begin{array}{ccc}8&4\\4&8\\\end{array}\right)=\left(\begin{array}{ccc}-3&0\\0&-3\\\end{array}\right)=-3\left(\begin{array}{ccc}1&0\\0&1\\\end{array}\right)=-3I_{2}$

$A+x\cdot I_{2}=\left(\begin{array}{ccc}2&1\\1&2\\\end{array}\right)+\left(\begin{array}{ccc}x&0\\0&x\\\end{array}\right)=\left(\begin{array}{ccc}2+x&1\\1&2+x\\\end{array}\right)$

$det(A+x\cdot I_{2})=\left|\begin{array}{ccc}2+x&1\\1&2+x\\\end{array}\right|=(2+x)^{2} -1$

(2+x)²-1=0

4+4x+x²-1=0

x²+4x+3=0

Δ=16-12=4

$x_{1}=\frac{-4+2}{2} =-1\\\\x_{2}=\frac{-4-2}{2} =-3$

$A^{-1} =\frac{1}{detA} \cdot A^{*}$

detA=4-1=3

$A^{t} = \left(\begin{array}{ccc}2&1\\1&2\\\end{array}\right)$

$A^{*} =\left(\begin{array}{ccc}2&-1\\-1&2\\\end{array}\right)$

$A^{-1} =\frac{1}{3} \cdot \left(\begin{array}{ccc}2&-1\\-1&2\\\end{array}\right)$