LA2.2.9

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Consider A_{{3\times 3}}=I_{{3\times 3}}A_{{3\times 3}}I_{{3\times 3}}\,

Therefore,{\begin{bmatrix}1&-1&-1\\1&1&1\\3&1&1\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}A{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\,

C_{2}-C_{1},C_{3}-C_{1}\, implies

{\begin{bmatrix}1&0&0\\1&2&2\\3&4&4\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}A{\begin{bmatrix}1&1&1\\0&1&0\\0&0&1\end{bmatrix}}\,

R_{2}-R_{1},R_{3}-3R_{1}\, implies

{\begin{bmatrix}1&0&0\\0&2&2\\0&4&4\end{bmatrix}}={\begin{bmatrix}1&0&0\\-1&1&0\\-3&0&1\end{bmatrix}}A{\begin{bmatrix}1&1&1\\0&1&0\\0&0&1\end{bmatrix}}\,

{\frac  {1}{2}}R_{2},{\frac  {1}{4}}R_{3}\, gives

{\begin{bmatrix}1&0&0\\0&1&1\\0&1&1\end{bmatrix}}={\begin{bmatrix}1&0&0\\-{\frac  {1}{2}}&{\frac  {1}{2}}&0\\-{\frac  {3}{4}}&0&{\frac  {1}{4}}\end{bmatrix}}A{\begin{bmatrix}1&1&1\\0&1&0\\0&0&1\end{bmatrix}}\,

R_{3}-R_{2}\, gives

{\begin{bmatrix}1&0&0\\0&1&1\\0&0&0\end{bmatrix}}={\begin{bmatrix}1&0&0\\-{\frac  {1}{2}}&{\frac  {1}{2}}&0\\-{\frac  {1}{4}}&-{\frac  {1}{2}}&{\frac  {1}{4}}\end{bmatrix}}A{\begin{bmatrix}1&1&1\\0&1&0\\0&0&1\end{bmatrix}}\,

C_{3}-C_{2}\, =

{\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}}={\begin{bmatrix}1&0&0\\-{\frac  {1}{2}}&{\frac  {1}{2}}&0\\-{\frac  {1}{4}}&-{\frac  {1}{2}}&{\frac  {1}{4}}\end{bmatrix}}A{\begin{bmatrix}1&1&0\\0&1&-1\\0&0&1\end{bmatrix}}\,

Thus the LHS is in the normal form {\begin{bmatrix}I_{2}&0\\0&0\end{bmatrix}}\,

Hence P_{{3\times 3}}={\begin{bmatrix}1&0&0\\-{\frac  {1}{2}}&{\frac  {1}{2}}&0\\-{\frac  {1}{4}}&-{\frac  {1}{2}}&{\frac  {1}{4}}\end{bmatrix}}\,

and Q_{{3\times 3}}={\begin{bmatrix}1&1&0\\0&1&-1\\0&0&1\end{bmatrix}}\,

Rank of A is equal to 2.

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