- Calcule as matrizes inversas das 4 matrizes abaixo. 4) $$ A=\left[\begin{array}{ll}3 & 5 \ 1 & 2\end{array} ight] $$ 5) $$ \mathrm{B}=\left[\begin{array}{rrr}-3 & 4 & -5 \ 0 & 1 & 2 \ 3 & -5 & 4\end{array} ight] $$ 6) $$ C=\left[\begin{array}{llll}1 & 0 & 0 & 0 \ 2 & 1 & 0 & 0 \ 3 & 2 & 1 & 0 \ 4 & 3 & 2 & 1\end{array} ight] $$ 7) $$ \mathrm{D}=\left[\begin{array}{rrr}1 & 0 & -2 \ -3 & -2 & -2 \ -3 & 0 & 2\end{array} ight] $$

Question

- Calcule as matrizes inversas das 4 matrizes abaixo. 4) $$ A=\left[\begin{array}{ll}3 & 5 \ 1 & 2\end{array}ight] $$ 5) $$ \mathrm{B}=\left[\begin{array}{rrr}-3 & 4 & -5 \ 0 & 1 & 2 \ 3 & -5 & 4\end{array}ight] $$ 6) $$ C=\left[\begin{array}{llll}1 & 0 & 0 & 0 \ 2 & 1 & 0 & 0 \ 3 & 2 & 1 & 0 \ 4 & 3 & 2 & 1\end{array}ight] $$ 7) $$ \mathrm{D}=\left[\begin{array}{rrr}1 & 0 & -2 \ -3 & -2 & -2 \ -3 & 0 & 2\end{array}ight] $$

UpStudy AI · Accepted Answer

Matrices by following steps:
- step0: Find the matrix inverse:
  $$\left[\begin{array}{ll}{3}&{5}\{1}&{2}\end{array}\right]$$
- step1: Evaluate the determinant:
  $$1$$
- step2: Using the formula:
  $$\frac{1}{3\times 2-5\times 1}\times \left[\begin{array}{ll}{2}&{-5}\{-1}&{3}\end{array}\right]$$
- step3: Evaluate:
  $$1\times \left[\begin{array}{ll}{2}&{-5}\{-1}&{3}\end{array}\right]$$
- step4: Calculate the product:
  $$\left[\begin{array}{ll}{2}&{-5}\{-1}&{3}\end{array}\right]$$
Find the inverse matrix of $$ \left[\begin{array}{rrr}-3 & 4 & -5 \ 0 & 1 & 2 \ 3 & -5 & 4\end{array}\right] $$.
Matrices by following steps:
- step0: Find the matrix inverse:
  $$\left[\begin{array}{lll}{-3}&{4}&{-5}\{0}&{1}&{2}\{3}&{-5}&{4}\end{array}\right]$$
- step1: Begin by adjoining the identity matrix to form the matrix:
  $$\left[\begin{array}{lll|lll}{-3}&{4}&{-5}&{1}&{0}&{0}\{0}&{1}&{2}&{0}&{1}&{0}\{3}&{-5}&{4}&{0}&{0}&{1}\end{array}\right]$$
- step2: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{4}{3}}&{\frac{5}{3}}&{-\frac{1}{3}}&{0}&{0}\{0}&{1}&{2}&{0}&{1}&{0}\{3}&{-5}&{4}&{0}&{0}&{1}\end{array}\right]$$
- step3: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{4}{3}}&{\frac{5}{3}}&{-\frac{1}{3}}&{0}&{0}\{0}&{1}&{2}&{0}&{1}&{0}\{0}&{-1}&{-1}&{1}&{0}&{1}\end{array}\right]$$
- step4: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{4}{3}}&{\frac{5}{3}}&{-\frac{1}{3}}&{0}&{0}\{0}&{1}&{2}&{0}&{1}&{0}\{0}&{0}&{1}&{1}&{1}&{1}\end{array}\right]$$
- step5: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{4}{3}}&{\frac{5}{3}}&{-\frac{1}{3}}&{0}&{0}\{0}&{1}&{0}&{-2}&{-1}&{-2}\{0}&{0}&{1}&{1}&{1}&{1}\end{array}\right]$$
- step6: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{4}{3}}&{0}&{-2}&{-\frac{5}{3}}&{-\frac{5}{3}}\{0}&{1}&{0}&{-2}&{-1}&{-2}\{0}&{0}&{1}&{1}&{1}&{1}\end{array}\right]$$
- step7: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{0}&{-\frac{14}{3}}&{-3}&{-\frac{13}{3}}\{0}&{1}&{0}&{-2}&{-1}&{-2}\{0}&{0}&{1}&{1}&{1}&{1}\end{array}\right]$$
- step8: Write the right half matrix:
  $$\left[\begin{array}{lll}{-\frac{14}{3}}&{-3}&{-\frac{13}{3}}\{-2}&{-1}&{-2}\{1}&{1}&{1}\end{array}\right]$$
Find the inverse matrix of $$ \left[\begin{array}{rrr}1 & 0 & -2 \ -3 & -2 & -2 \ -3 & 0 & 2\end{array}\right] $$.
Matrices by following steps:
- step0: Find the matrix inverse:
  $$\left[\begin{array}{lll}{1}&{0}&{-2}\{-3}&{-2}&{-2}\{-3}&{0}&{2}\end{array}\right]$$
- step1: Begin by adjoining the identity matrix to form the matrix:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{-2}&{1}&{0}&{0}\{-3}&{-2}&{-2}&{0}&{1}&{0}\{-3}&{0}&{2}&{0}&{0}&{1}\end{array}\right]$$
- step2: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{-2}&{1}&{0}&{0}\{0}&{-2}&{-8}&{3}&{1}&{0}\{-3}&{0}&{2}&{0}&{0}&{1}\end{array}\right]$$
- step3: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{-2}&{1}&{0}&{0}\{0}&{-2}&{-8}&{3}&{1}&{0}\{0}&{0}&{-4}&{3}&{0}&{1}\end{array}\right]$$
- step4: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{-2}&{1}&{0}&{0}\{0}&{1}&{4}&{-\frac{3}{2}}&{-\frac{1}{2}}&{0}\{0}&{0}&{-4}&{3}&{0}&{1}\end{array}\right]$$
- step5: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{-2}&{1}&{0}&{0}\{0}&{1}&{4}&{-\frac{3}{2}}&{-\frac{1}{2}}&{0}\{0}&{0}&{1}&{-\frac{3}{4}}&{0}&{-\frac{1}{4}}\end{array}\right]$$
- step6: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{-2}&{1}&{0}&{0}\{0}&{1}&{0}&{\frac{3}{2}}&{-\frac{1}{2}}&{1}\{0}&{0}&{1}&{-\frac{3}{4}}&{0}&{-\frac{1}{4}}\end{array}\right]$$
- step7: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{0}&{-\frac{1}{2}}&{0}&{-\frac{1}{2}}\{0}&{1}&{0}&{\frac{3}{2}}&{-\frac{1}{2}}&{1}\{0}&{0}&{1}&{-\frac{3}{4}}&{0}&{-\frac{1}{4}}\end{array}\right]$$
- step8: Write the right half matrix:
  $$\left[\begin{array}{lll}{-\frac{1}{2}}&{0}&{-\frac{1}{2}}\{\frac{3}{2}}&{-\frac{1}{2}}&{1}\{-\frac{3}{4}}&{0}&{-\frac{1}{4}}\end{array}\right]$$
Find the inverse matrix of $$ \left[\begin{array}{llll}1 & 0 & 0 & 0 \ 2 & 1 & 0 & 0 \ 3 & 2 & 1 & 0 \ 4 & 3 & 2 & 1\end{array}\right] $$.
Matrices by following steps:
- step0: Find the matrix inverse:
  $$\left[\begin{array}{llll}{1}&{0}&{0}&{0}\{2}&{1}&{0}&{0}\{3}&{2}&{1}&{0}\{4}&{3}&{2}&{1}\end{array}\right]$$
- step1: Begin by adjoining the identity matrix to form the matrix:
  $$\left[\begin{array}{llll|llll}{1}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\{2}&{1}&{0}&{0}&{0}&{1}&{0}&{0}\{3}&{2}&{1}&{0}&{0}&{0}&{1}&{0}\{4}&{3}&{2}&{1}&{0}&{0}&{0}&{1}\end{array}\right]$$
- step2: Simplify the row:
  $$\left[\begin{array}{llll|llll}{1}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\{3}&{2}&{1}&{0}&{0}&{0}&{1}&{0}\{4}&{3}&{2}&{1}&{0}&{0}&{0}&{1}\end{array}\right]$$
- step3: Simplify the row:
  $$\left[\begin{array}{llll|llll}{1}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\{0}&{2}&{1}&{0}&{-3}&{0}&{1}&{0}\{4}&{3}&{2}&{1}&{0}&{0}&{0}&{1}\end{array}\right]$$
- step4: Simplify the row:
  $$\left[\begin{array}{llll|llll}{1}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\{0}&{2}&{1}&{0}&{-3}&{0}&{1}&{0}\{0}&{3}&{2}&{1}&{-4}&{0}&{0}&{1}\end{array}\right]$$
- step5: Simplify the row:
  $$\left[\begin{array}{llll|llll}{1}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\{0}&{0}&{1}&{0}&{1}&{-2}&{1}&{0}\{0}&{3}&{2}&{1}&{-4}&{0}&{0}&{1}\end{array}\right]$$
- step6: Simplify the row:
  $$\left[\begin{array}{llll|llll}{1}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\{0}&{0}&{1}&{0}&{1}&{-2}&{1}&{0}\{0}&{0}&{2}&{1}&{2}&{-3}&{0}&{1}\end{array}\right]$$
- step7: Simplify the row:
  $$\left[\begin{array}{llll|llll}{1}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\{0}&{1}&{0}&{0}&{-2}&{1}&{0}&{0}\{0}&{0}&{1}&{0}&{1}&{-2}&{1}&{0}\{0}&{0}&{0}&{1}&{0}&{1}&{-2}&{1}\end{array}\right]$$
- step8: Write the right half matrix:
  $$\left[\begin{array}{llll}{1}&{0}&{0}&{0}\{-2}&{1}&{0}&{0}\{1}&{-2}&{1}&{0}\{0}&{1}&{-2}&{1}\end{array}\right]$$
Aqui estão as matrizes inversas das quatro matrizes fornecidas:

4) A inversa da matriz $$ A = \left[\begin{array}{ll}3 & 5 \ 1 & 2\end{array}\right] $$ é:
$$
A^{-1} = \left[\begin{array}{ll}2 & -5 \ -1 & 3\end{array}\right]
$$

5) A inversa da matriz $$ B = \left[\begin{array}{rrr}-3 & 4 & -5 \ 0 & 1 & 2 \ 3 & -5 & 4\end{array}\right] $$ é:
$$
B^{-1} = \left[\begin{array}{lll}-\frac{14}{3} & -3 & -\frac{13}{3} \ -2 & -1 & -2 \ 1 & 1 & 1\end{array}\right]
$$

6) A inversa da matriz $$ C = \left[\begin{array}{llll}1 & 0 & 0 & 0 \ 2 & 1 & 0 & 0 \ 3 & 2 & 1 & 0 \ 4 & 3 & 2 & 1\end{array}\right] $$ é:
$$
C^{-1} = \left[\begin{array}{llll}1 & 0 & 0 & 0 \ -2 & 1 & 0 & 0 \ 1 & -2 & 1 & 0 \ 0 & 1 & -2 & 1\end{array}\right]
$$

7) A inversa da matriz $$ D = \left[\begin{array}{rrr}1 & 0 & -2 \ -3 & -2 & -2 \ -3 & 0 & 2\end{array}\right] $$ é:
$$
D^{-1} = \left[\begin{array}{lll}-\frac{1}{2} & 0 & -\frac{1}{2} \ \frac{3}{2} & -\frac{1}{2} & 1 \ -\frac{3}{4} & 0 & -\frac{1}{4}\end{array}\right]
$$
As matrizes inversas são: 4) $$ A^{-1} = \left[\begin{array}{ll}2 & -5 \ -1 & 3\end{array}\right] $$ 5) $$ B^{-1} = \left[\begin{array}{lll}-\frac{14}{3} & -3 & -\frac{13}{3} \ -2 & -1 & -2 \ 1 & 1 & 1\end{array}\right] $$ 6) $$ C^{-1} = \left[\begin{array}{llll}1 & 0 & 0 & 0 \ -2 & 1 & 0 & 0 \ 1 & -2 & 1 & 0 \ 0 & 1 & -2 & 1\end{array}\right] $$ 7) $$ D^{-1} = \left[\begin{array}{lll}-\frac{1}{2} & 0 & -\frac{1}{2} \ \frac{3}{2} & -\frac{1}{2} & 1 \ -\frac{3}{4} & 0 & -\frac{1}{4}\end{array}\right] $$

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