LA2.2.8

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Consider A_{{3\times 4}}=I_{{3\times 3}}A_{{3\times 4}}I_{{4\times 4}}\,

{\begin{bmatrix}1&3&6&-1\\1&4&5&1\\1&5&4&3\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}AI_{4}\,

Applying R_{2}-R_{1},R_{3}-R_{1}\,,we get

{\begin{bmatrix}1&3&6&-1\\0&1&-1&2\\0&2&-2&4\end{bmatrix}}={\begin{bmatrix}1&0&0\\-1&1&0\\-1&0&1\end{bmatrix}}AI_{4}\,

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

{\begin{bmatrix}1&3&6&-1\\0&1&-1&2\\0&0&0&0\end{bmatrix}}={\begin{bmatrix}1&0&0\\-1&1&0\\1&-2&1\end{bmatrix}}AI_{4}\,

Applying C_{2}-3C_{1},C_{3}-6C_{1},C_{4}-C_{1}\, to the above matrix,we get

{\begin{bmatrix}1&0&0&0\\0&1&-1&2\\0&0&0&0\end{bmatrix}}={\begin{bmatrix}1&0&0\\-1&1&0\\1&-2&1\end{bmatrix}}\times A{\begin{bmatrix}1&-3&-6&1\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\,

Now C_{3}-C_{2},C_{4}-2C_{2}\, implies

{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}={\begin{bmatrix}1&0&0\\-1&1&0\\1&-2&1\end{bmatrix}}\times A{\begin{bmatrix}1&-3&-9&7\\0&1&1&-2\\0&0&1&0\\0&0&0&1\end{bmatrix}}\,

Thus I_{2}=PAQ\, where P={\begin{bmatrix}1&0&0\\-1&1&0\\1&-2&1\end{bmatrix}}\,

Q={\begin{bmatrix}1&-3&-9&7\\0&1&1&-2\\0&0&1&0\\0&0&0&1\end{bmatrix}}\,

Rank of A is 2.

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