Practice set five

Find the inverse of every following matrices


1). \(\left[\begin{array}{ccc}1 & 0 & -1 \\ 2 & 0 & 1 \\ 1 & 1&1\end{array}\right]\) 


2). \(\left[\begin{array}{ccc}8 & 0 & 1 \\ 0 & 2 & 0 \\ 0 & 0 & -1\end{array}\right]\) 


3). \(\left[\begin{array}{ccc}-1 & 2 & 3 \\ 0 & 4 & 3 \\ -1 & 2 & 3\end{array}\right]\)


4). \(\left[\begin{array}{ccc}1 & 3 & 1 \\ 1 & 4&2 \\ 0 & 2 & -2\end{array}\right]\) 


5). \(\left[\begin{array}{lll}3 & 6 & 1 \\ 4 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]\)




Answer:

1). \(\left[\begin{array}{ccc}\frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{-2}{3} & 1 \\ -\frac{2}{3} & \frac{1}{3} & 0\end{array}\right]\) 


2). \(\left[\begin{array}{ccc}\frac{1}{8} & 0 & \frac{1}{8} \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & -1\end{array}\right]\) 


3). Matrix is singular ,so it has no inverse


4). \(\left[\begin{array}{ccc}3 & -2 & -\frac{1}{2} \\ \frac{-1}{2} & \frac{1}{2} & \frac{1}{4} \\ \frac{-1}{2} & \frac{1}{2} & \frac{-1}{4}\end{array}\right]\)


5). \(\left[\begin{array}{ccc}\frac{-1}{21} & \frac{2}{7} & -\frac{5}{21} \\ \frac{4}{21} & \frac{-1}{7} & \frac{-1}{21} \\ 0 & 0 & 1\end{array}\right]\)

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