Calcular inversa

Ejercicios_Resueltos matrices matriz inversa

Calcula la inversa de la matriz A
$A = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 3 & 1 & 1 \end{array} \right)$

SOLUCIÓN

Matriz inversa

$A = \begin{pmatrix}1 & 0 & 0 \\ 4 & 1 & 0 \\ 3 & 1 & 1\end{pmatrix}$

$\det(A) = 1 \neq 0 \implies \exists\, A^{-1}$

Método de Gauss-Jordan

Matriz ampliada A :

$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & 1 & 0 & 0 \\ 4 & 1 & 0 & 0 & 1 & 0 \\ 3 & 1 & 1 & 0 & 0 & 1\end{array}\right]$

$F_2 = F_2 - 4\cdot F_1$

$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -4 & 1 & 0 \\ 3 & 1 & 1 & 0 & 0 & 1\end{array}\right]$

$F_3 = F_3 - 3\cdot F_1$

$\left[\begin{array}{ccc|ccc}1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -4 & 1 & 0 \\ 0 & 1 & 1 & -3 & 0 & 1\end{array}\right]$

$F_3 = F_3 - F_2$

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

$A^{-1} = \begin{pmatrix}1 & 0 & 0 \\ -4 & 1 & 0 \\ 1 & -1 & 1\end{pmatrix}$

Usando determinantes: A⁻¹ = (1/|A|) · Adj(A)

$A^{-1}=\dfrac{1}{|A|}\cdot\left(\text{Adj}\,A\right)^t$

Paso 1 · Determinante de A:

$|A| = 1$

Paso 2 · Cofactores:

$C_{11}=(-1)^{1+1}\cdot\left|\begin{array}{cc}1 & 0 \\ 1 & 1\end{array}\right|=(+1)\cdot1=1$

$C_{12}=(-1)^{1+2}\cdot\left|\begin{array}{cc}4 & 0 \\ 3 & 1\end{array}\right|=(-1)\cdot4=-4$

$C_{13}=(-1)^{1+3}\cdot\left|\begin{array}{cc}4 & 1 \\ 3 & 1\end{array}\right|=(+1)\cdot1=1$

$C_{21}=(-1)^{2+1}\cdot\left|\begin{array}{cc}0 & 0 \\ 1 & 1\end{array}\right|=(-1)\cdot0=0$

$C_{22}=(-1)^{2+2}\cdot\left|\begin{array}{cc}1 & 0 \\ 3 & 1\end{array}\right|=(+1)\cdot1=1$

$C_{23}=(-1)^{2+3}\cdot\left|\begin{array}{cc}1 & 0 \\ 3 & 1\end{array}\right|=(-1)\cdot1=-1$

$C_{31}=(-1)^{3+1}\cdot\left|\begin{array}{cc}0 & 0 \\ 1 & 0\end{array}\right|=(+1)\cdot0=0$

$C_{32}=(-1)^{3+2}\cdot\left|\begin{array}{cc}1 & 0 \\ 4 & 0\end{array}\right|=(-1)\cdot0=0$

$C_{33}=(-1)^{3+3}\cdot\left|\begin{array}{cc}1 & 0 \\ 4 & 1\end{array}\right|=(+1)\cdot1=1$

Paso 3 · Matriz de cofactores:

$C = \begin{pmatrix}1 & -4 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{pmatrix}$

Paso 4 · Adjunta (traspuesta de C):

$\text{Adj}\,A = C^t = \begin{pmatrix}1 & 0 & 0 \\ -4 & 1 & 0 \\ 1 & -1 & 1\end{pmatrix}$

Paso 5 · Inversa:

$A^{-1}=\dfrac{1}{1}\begin{pmatrix}1 & 0 & 0 \\ -4 & 1 & 0 \\ 1 & -1 & 1\end{pmatrix}=\begin{pmatrix}1 & 0 & 0 \\ -4 & 1 & 0 \\ 1 & -1 & 1\end{pmatrix}$

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