5. Tind the inverse of the matrix $$ A=\left[\begin{array}{ccc}4 & -2 & 1 \ 7 & 3 & 3 \ 2 & 0 & 1\end{array} ight] $$

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5. Tind the inverse of the matrix $$ A=\left[\begin{array}{ccc}4 & -2 & 1 \ 7 & 3 & 3 \ 2 & 0 & 1\end{array}ight] $$

UpStudy AI · Accepted Answer

Matrices by following steps:
- step0: Find the matrix inverse:
  $$\left[\begin{array}{lll}{4}&{-2}&{1}\{7}&{3}&{3}\{2}&{0}&{1}\end{array}\right]$$
- step1: Begin by adjoining the identity matrix to form the matrix:
  $$\left[\begin{array}{lll|lll}{4}&{-2}&{1}&{1}&{0}&{0}\{7}&{3}&{3}&{0}&{1}&{0}\{2}&{0}&{1}&{0}&{0}&{1}\end{array}\right]$$
- step2: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{7}&{3}&{3}&{0}&{1}&{0}\{2}&{0}&{1}&{0}&{0}&{1}\end{array}\right]$$
- step3: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{\frac{13}{2}}&{\frac{5}{4}}&{-\frac{7}{4}}&{1}&{0}\{2}&{0}&{1}&{0}&{0}&{1}\end{array}\right]$$
- step4: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{\frac{13}{2}}&{\frac{5}{4}}&{-\frac{7}{4}}&{1}&{0}\{0}&{1}&{\frac{1}{2}}&{-\frac{1}{2}}&{0}&{1}\end{array}\right]$$
- step5: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{1}&{\frac{5}{26}}&{-\frac{7}{26}}&{\frac{2}{13}}&{0}\{0}&{1}&{\frac{1}{2}}&{-\frac{1}{2}}&{0}&{1}\end{array}\right]$$
- step6: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{1}&{\frac{5}{26}}&{-\frac{7}{26}}&{\frac{2}{13}}&{0}\{0}&{0}&{\frac{4}{13}}&{-\frac{3}{13}}&{-\frac{2}{13}}&{1}\end{array}\right]$$
- step7: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{1}&{\frac{5}{26}}&{-\frac{7}{26}}&{\frac{2}{13}}&{0}\{0}&{0}&{1}&{-\frac{3}{4}}&{-\frac{1}{2}}&{\frac{13}{4}}\end{array}\right]$$
- step8: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{1}{2}}&{\frac{1}{4}}&{\frac{1}{4}}&{0}&{0}\{0}&{1}&{0}&{-\frac{1}{8}}&{\frac{1}{4}}&{-\frac{5}{8}}\{0}&{0}&{1}&{-\frac{3}{4}}&{-\frac{1}{2}}&{\frac{13}{4}}\end{array}\right]$$
- step9: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{-\frac{1}{2}}&{0}&{\frac{7}{16}}&{\frac{1}{8}}&{-\frac{13}{16}}\{0}&{1}&{0}&{-\frac{1}{8}}&{\frac{1}{4}}&{-\frac{5}{8}}\{0}&{0}&{1}&{-\frac{3}{4}}&{-\frac{1}{2}}&{\frac{13}{4}}\end{array}\right]$$
- step10: Simplify the row:
  $$\left[\begin{array}{lll|lll}{1}&{0}&{0}&{\frac{3}{8}}&{\frac{1}{4}}&{-\frac{9}{8}}\{0}&{1}&{0}&{-\frac{1}{8}}&{\frac{1}{4}}&{-\frac{5}{8}}\{0}&{0}&{1}&{-\frac{3}{4}}&{-\frac{1}{2}}&{\frac{13}{4}}\end{array}\right]$$
- step11: Write the right half matrix:
  $$\left[\begin{array}{lll}{\frac{3}{8}}&{\frac{1}{4}}&{-\frac{9}{8}}\{-\frac{1}{8}}&{\frac{1}{4}}&{-\frac{5}{8}}\{-\frac{3}{4}}&{-\frac{1}{2}}&{\frac{13}{4}}\end{array}\right]$$
The inverse of the matrix

$$
A = \left[\begin{array}{ccc} 4 & -2 & 1 \ 7 & 3 & 3 \ 2 & 0 & 1 \end{array}\right]
$$

is

$$
A^{-1} = \left[\begin{array}{ccc} \frac{3}{8} & \frac{1}{4} & -\frac{9}{8} \ -\frac{1}{8} & \frac{1}{4} & -\frac{5}{8} \ -\frac{3}{4} & -\frac{1}{2} & \frac{13}{4} \end{array}\right].
$$
The inverse of matrix $$ A $$ is $$ A^{-1} = \left[\begin{array}{ccc} \frac{3}{8} & \frac{1}{4} & -\frac{9}{8} \ -\frac{1}{8} & \frac{1}{4} & -\frac{5}{8} \ -\frac{3}{4} & -\frac{1}{2} & \frac{13}{4} \end{array}\right]. $$

5. Tind the inverse of the matrix \( A=\left[\begin{array}{ccc}4 & -2 & 1 \\ 7 & 3 & 3 \\ 2 & 0 & 1\end{array}\right] \)

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