Dada la matriz B: $$ B=\left(\begin{array}{lll}4 & 2 & 1 \ 3 & 3 & 1 \ 2 & 1 & 2\end{array} ight) $$ Calcule la matriz inversa

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Dada la matriz B: $$ B=\left(\begin{array}{lll}4 & 2 & 1 \ 3 & 3 & 1 \ 2 & 1 & 2\end{array}ight) $$ Calcule la matriz inversa

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Matrices by following steps:
- step0: Find the matrix inverse:
  $$\left[\begin{array}{lll}{4}&{2}&{1}\{3}&{3}&{1}\{2}&{1}&{2}\end{array}\right]$$
- step1: Begin by adjoining the identity matrix to form the matrix:
  $$\left[\begin{array}{lll|lll}{4}&{2}&{1}&{1}&{0}&{0}\{3}&{3}&{1}&{0}&{1}&{0}\{2}&{1}&{2}&{0}&{0}&{1}\end{array}\right]$$
- step2: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{3}&{3}&{1}&{0}&{1}&{0}\{2}&{1}&{2}&{0}&{0}&{1}\end{array}\right]$$
- step3: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{\frac{3}{2}}&{\frac{1}{4}}&{-\frac{3}{4}}&{1}&{0}\{2}&{1}&{2}&{0}&{0}&{1}\end{array}\right]$$
- step4: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{\frac{3}{2}}&{\frac{1}{4}}&{-\frac{3}{4}}&{1}&{0}\{0}&{0}&{\frac{3}{2}}&{-\frac{1}{2}}&{0}&{1}\end{array}\right]$$
- step5: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{1}&{\frac{1}{6}}&{-\frac{1}{2}}&{\frac{2}{3}}&{0}\{0}&{0}&{\frac{3}{2}}&{-\frac{1}{2}}&{0}&{1}\end{array}\right]$$
- step6: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{1}&{\frac{1}{6}}&{-\frac{1}{2}}&{\frac{2}{3}}&{0}\{0}&{0}&{1}&{-\frac{1}{3}}&{0}&{\frac{2}{3}}\end{array}\right]$$
- step7: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{1}&{0}&{-\frac{4}{9}}&{\frac{2}{3}}&{-\frac{1}{9}}\{0}&{0}&{1}&{-\frac{1}{3}}&{0}&{\frac{2}{3}}\end{array}\right]$$
- step8: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{\frac{1}{2}}&{0}&{\frac{1}{3}}&{0}&{-\frac{1}{6}}\{0}&{1}&{0}&{-\frac{4}{9}}&{\frac{2}{3}}&{-\frac{1}{9}}\{0}&{0}&{1}&{-\frac{1}{3}}&{0}&{\frac{2}{3}}\end{array}\right]$$
- step9: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{0}&{\frac{5}{9}}&{-\frac{1}{3}}&{-\frac{1}{9}}\{0}&{1}&{0}&{-\frac{4}{9}}&{\frac{2}{3}}&{-\frac{1}{9}}\{0}&{0}&{1}&{-\frac{1}{3}}&{0}&{\frac{2}{3}}\end{array}\right]$$
- step10: Write the right half matrix:
  $$\left[\begin{array}{lll}{\frac{5}{9}}&{-\frac{1}{3}}&{-\frac{1}{9}}\{-\frac{4}{9}}&{\frac{2}{3}}&{-\frac{1}{9}}\{-\frac{1}{3}}&{0}&{\frac{2}{3}}\end{array}\right]$$
La matriz inversa de $$ B $$ es:

$$
B^{-1} = \left[ \begin{array}{lll}
\frac{5}{9} & -\frac{1}{3} & -\frac{1}{9} \
-\frac{4}{9} & \frac{2}{3} & -\frac{1}{9} \
-\frac{1}{3} & 0 & \frac{2}{3}
\end{array} \right]
$$
La matriz inversa de $$ B $$ es: $$ B^{-1} = \left[ \begin{array}{ccc} \frac{5}{9} & -\frac{1}{3} & -\frac{1}{9} \ -\frac{4}{9} & \frac{2}{3} & -\frac{1}{9} \ -\frac{1}{3} & 0 & \frac{2}{3} \end{array} \right] $$

Dada la matriz B: \( B=\left(\begin{array}{lll}4 & 2 & 1 \\ 3 & 3 & 1 \\ 2 & 1 & 2\end{array}\right) \) Calcule la matriz inversa

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