Write the augmented matrix for the homogeneous system \(\left( {P - I} \right){\mathop{\rm x}\nolimits} = 0\)as shown below:
\(\left( {\begin{array}{*{20}{c}}{ - .3}&{.1}&{.1}&0\\{.2}&{ - .2}&{.2}&0\\{.1}&{.1}&{ - .3}&0\end{array}} \right)\)
Perform an elementary row operation to produce the row-reduced echelon form of the matrix.
At row 1, multiply row 1 by \( - \frac{1}{{0.3}}\).
\( \sim \left( {\begin{array}{*{20}{c}}1&{ - .333}&{ - .3333}&0\\{.2}&{ - .2}&{.2}&0\\{.1}&{.1}&{ - .3}&0\end{array}} \right)\)
At row 2, multiply row 1 by 0.2 and subtract it from row 2. At row 3, multiply row 1 by 0.1 and subtract it from row 3.
\( \sim \left( {\begin{array}{*{20}{c}}1&{ - .333}&{ - .3333}&0\\0&{ - 0.1333}&{0.2666}&0\\0&{0.1333}&{0.2666}&0\end{array}} \right)\)
At row 2, multiply row 2 by \( - \frac{1}{{0.1333}}\).
\( \sim \left( {\begin{array}{*{20}{c}}1&{ - .333}&{ - .3333}&0\\0&1&{ - 2}&0\\0&{0.1333}&{0.2666}&0\end{array}} \right)\)
At row 1, multiply row 2 by 0.333 and add it to row 1. At row 3, multiply row 2 by 0.1333 and subtract it from row 3.
\( \sim \left( {\begin{array}{*{20}{c}}1&0&{ - 1}&0\\0&1&{ - 2}&0\\0&0&0&0\end{array}} \right)\)