Question
upstudy study bank question image url

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

Ask by Osborne Clarke. in Mexico
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

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

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Beyond the Answer

Para calcular la matriz inversa de \( B \), podemos usar la fórmula \( B^{-1} = \frac{1}{\text{det}(B)} \cdot \text{adj}(B) \), donde \( \text{det}(B) \) es el determinante de la matriz y \( \text{adj}(B) \) es la matriz adjunta. Primero, encontramos el determinante: \[ \text{det}(B) = 4(3 \cdot 2 - 1 \cdot 1) - 2(3 \cdot 2 - 1 \cdot 2) + 1(3 \cdot 1 - 3 \cdot 2) \] Calculando esto, obtenemos \( \text{det}(B) = 4(6 - 1) - 2(6 - 2) + 1(3 - 6) = 4(5) - 2(4) - 3 = 20 - 8 - 3 = 9 \). Luego, calculamos la matriz adjunta. La matriz de cofactores se puede obtener calculando los menores y aplicando el signo correspondiente: Esos valores nos dan la matriz adjunta \( \text{adj}(B) \) y luego podemos calcular la inversa. Finalmente, la matriz inversa será: \[ B^{-1} = \frac{1}{9} \cdot \text{adj}(B) \] Por lo tanto, la matriz inversa \( B^{-1} \) será: \[ B^{-1} = \frac{1}{9}\begin{pmatrix}7 & -8 & -1 \\-3 & 13 & -6 \\ 1 & -6 & 3\end{pmatrix} = \begin{pmatrix}\frac{7}{9} & -\frac{8}{9} & -\frac{1}{9} \\ -\frac{1}{3} & \frac{13}{9} & -\frac{2}{3} \\ \frac{1}{9} & -\frac{2}{3} & \frac{1}{3}\end{pmatrix}. \]

Related Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy