Matrices Selectividad Castilla-La Mancha

castilla_mancha ecuacion_matricial Ejercicios_Resueltos matrices selectividad

– Despeja la matriz $X$ en la ecuación $A \cdot X - A = I - A \cdot X$

– Halla la matriz $X$ sabiendo que
$A = \left( \begin{array}{ccc} 1 & 1 & 0\\ 0 & 1 & 2\\ 1 & 0 & 1 \end{array} \right)$
e
$I = \left( \begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$

SOLUCIÓN

Matrices:

$A = \begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 2 \\ 1 & 0 & 1\end{pmatrix}\qquad I = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$

Ecuación:

$AX-A = I-AX$

Despejamos $X$ :

$(A+A)\cdot X = A+I$

Multiplicamos a la izquierda por $(A+A)^{-1}$ en ambos miembros:

$(A+A)^{-1}\cdot (A+A)\cdot X = (A+A)^{-1}\cdot (A+I)$

$I\cdot X = (A+A)^{-1}\cdot (A+I)$

$\boxed{X = (A+A)^{-1}\cdot (A+I)}$

Calculamos numéricamente:

Calculamos $A+I$ :

$A+I = \begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 2 \\ 1 & 0 & 1\end{pmatrix}+\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix} = \begin{pmatrix}2 & 1 & 0 \\ 0 & 2 & 2 \\ 1 & 0 & 2\end{pmatrix}$

Suma de coeficientes de $X$ :

Calculamos $(A+A)$ :

$(A+A) = \begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 2 \\ 1 & 0 & 1\end{pmatrix}+\begin{pmatrix}1 & 1 & 0 \\ 0 & 1 & 2 \\ 1 & 0 & 1\end{pmatrix} = \begin{pmatrix}2 & 2 & 0 \\ 0 & 2 & 4 \\ 2 & 0 & 2\end{pmatrix}$

Calculamos $(A+A)^{-1}$ (Gauss-Jordan):

Matriz ampliada $[M|I]$ :

$\left[\begin{array}{ccc|ccc}2 & 2 & 0 & 1 & 0 & 0 \\ 0 & 2 & 4 & 0 & 1 & 0 \\ 2 & 0 & 2 & 0 & 0 & 1\end{array}\right]$

$F_1=\dfrac{1}{2}\cdot F_1$

$\left[\begin{array}{ccc|ccc}1 & 1 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 2 & 4 & 0 & 1 & 0 \\ 2 & 0 & 2 & 0 & 0 & 1\end{array}\right]$

$F_3=F_3-2\cdot F_1$

$\left[\begin{array}{ccc|ccc}1 & 1 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 2 & 4 & 0 & 1 & 0 \\ 0 & -2 & 2 & -1 & 0 & 1\end{array}\right]$

$F_2=\dfrac{1}{2}\cdot F_2$

$\left[\begin{array}{ccc|ccc}1 & 1 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 1 & 2 & 0 & \frac{1}{2} & 0 \\ 0 & -2 & 2 & -1 & 0 & 1\end{array}\right]$

$F_1=F_1-F_2$

$\left[\begin{array}{ccc|ccc}1 & 0 & -2 & \frac{1}{2} & -\frac{1}{2} & 0 \\ 0 & 1 & 2 & 0 & \frac{1}{2} & 0 \\ 0 & -2 & 2 & -1 & 0 & 1\end{array}\right]$

$F_3=F_3+2\cdot F_2$

$\left[\begin{array}{ccc|ccc}1 & 0 & -2 & \frac{1}{2} & -\frac{1}{2} & 0 \\ 0 & 1 & 2 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 6 & -1 & 1 & 1\end{array}\right]$

$F_3=\dfrac{1}{6}\cdot F_3$

$\left[\begin{array}{ccc|ccc}1 & 0 & -2 & \frac{1}{2} & -\frac{1}{2} & 0 \\ 0 & 1 & 2 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1 & -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{array}\right]$

$F_1=F_1+2\cdot F_3$

$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & \frac{1}{6} & -\frac{1}{6} & \frac{1}{3} \\ 0 & 1 & 2 & 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1 & -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{array}\right]$

$F_2=F_2-2\cdot F_3$

$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & \frac{1}{6} & -\frac{1}{6} & \frac{1}{3} \\ 0 & 1 & 0 & \frac{1}{3} & \frac{1}{6} & -\frac{1}{3} \\ 0 & 0 & 1 & -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{array}\right]$

$(A+A)^{-1} = \begin{pmatrix}\frac{1}{6} & -\frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{6} & -\frac{1}{3} \\ -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{pmatrix}$

Calculamos $X = (A+A)^{-1}\cdot (A+I)$ :

$X = \begin{pmatrix}\frac{1}{6} & -\frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{6} & -\frac{1}{3} \\ -\frac{1}{6} & \frac{1}{6} & \frac{1}{6}\end{pmatrix}\begin{pmatrix}2 & 1 & 0 \\ 0 & 2 & 2 \\ 1 & 0 & 2\end{pmatrix}$

$X_{11} = \frac{1}{6}\cdot 2 + (-\frac{1}{6})\cdot 0 + \frac{1}{3}\cdot 1 = {\color{blue}\frac{2}{3}}$

$X_{12} = \frac{1}{6}\cdot 1 + (-\frac{1}{6})\cdot 2 + \frac{1}{3}\cdot 0 = {\color{blue}-\frac{1}{6}}$

$X_{13} = \frac{1}{6}\cdot 0 + (-\frac{1}{6})\cdot 2 + \frac{1}{3}\cdot 2 = {\color{blue}\frac{1}{3}}$

$X_{21} = \frac{1}{3}\cdot 2 + \frac{1}{6}\cdot 0 + (-\frac{1}{3})\cdot 1 = {\color{blue}\frac{1}{3}}$

$X_{22} = \frac{1}{3}\cdot 1 + \frac{1}{6}\cdot 2 + (-\frac{1}{3})\cdot 0 = {\color{blue}\frac{2}{3}}$

$X_{23} = \frac{1}{3}\cdot 0 + \frac{1}{6}\cdot 2 + (-\frac{1}{3})\cdot 2 = {\color{blue}-\frac{1}{3}}$

$X_{31} = (-\frac{1}{6})\cdot 2 + \frac{1}{6}\cdot 0 + \frac{1}{6}\cdot 1 = {\color{blue}-\frac{1}{6}}$

$X_{32} = (-\frac{1}{6})\cdot 1 + \frac{1}{6}\cdot 2 + \frac{1}{6}\cdot 0 = {\color{blue}\frac{1}{6}}$

$X_{33} = (-\frac{1}{6})\cdot 0 + \frac{1}{6}\cdot 2 + \frac{1}{6}\cdot 2 = {\color{blue}\frac{2}{3}}$

$X = \begin{pmatrix}\frac{2}{3} & -\frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{2}{3} & -\frac{1}{3} \\ -\frac{1}{6} & \frac{1}{6} & \frac{2}{3}\end{pmatrix}$

Resultado:

$\boxed{X = \begin{pmatrix}\frac{2}{3} & -\frac{1}{6} & \frac{1}{3} \\ \frac{1}{3} & \frac{2}{3} & -\frac{1}{3} \\ -\frac{1}{6} & \frac{1}{6} & \frac{2}{3}\end{pmatrix}}$

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