Question

Un sistema de control automático se representa con la ecuación matricialHalla el valor de X

#Ser Bachiller

Answer 1

Dado la ecuación del sistema el valor de X es:

$X = \left[\begin{array}{cc}2&1\\-2&-10\end{array}\right]$

Explicación:

Datos;

$A=\left[\begin{array}{cc}1&0\\3&1\end{array}\right]$

Calcular A²;

$A^{2} =\left[\begin{array}{cc}1&0\\3&1\end{array}\right] .\left[\begin{array}{cc}1&0\\3&1\end{array}\right]$

$A^{2}= \left[\begin{array}{cc}1&0\\6&1\end{array}\right]$

$B=\left[\begin{array}{cc}-2&-1\\-10&4\end{array}\right]$

$-B=\left[\begin{array}{cc}2&1\\10&-4\end{array}\right]$

A²X + B = 0

A²X = -B

Si el det(A²) ≠ 0

det(A²) = 1(1)-0(3) = 1

Por o tanto;

X = -(A²)⁻¹B

Calcular (A²)⁻¹;

Aplicar formula matriz inversa;

$\left[\begin{array}{cc}a&b\\c&d\end{array}\right] = \frac{1}{det(\left[\begin{array}{cc}a&b\\c&d\end{array}\right])} = \left[\begin{array}{cc}d&-b\\-c&a\end{array}\right]$

Sustituir;

$(A^{2})^{-1} = \left[\begin{array}{cc}1&0\\-6&1\end{array}\right]$

Hallar x;

$X = \left[\begin{array}{cc}1&0\\-6&1\end{array}\right] \left[\begin{array}{cc}2&1\\10&-4\end{array}\right]$

$X = \left[\begin{array}{cc}2&1\\-2&-10\end{array}\right]$