LA2.2.7

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A={\begin{bmatrix}1&1&-1&0\\4&4&-3&1\\b&2&2&2\\9&9&b&3\end{bmatrix}}\,

Applying R_{2}-4R_{1},R_{3}-2R_{1},R_{4}-9R_{1}\, to the above matrix,we get

{\begin{bmatrix}1&1&-1&0\\0&0&1&1\\b-2&0&4&2\\0&0&b+9&3\end{bmatrix}}\,

Applying R_{3}-4R_{2},R_{4}-3R_{2}\, to the above,we get

{\begin{bmatrix}1&1&-1&0\\0&0&1&1\\b-2&0&0&-2\\0&0&b+6&0\end{bmatrix}}\,

Now R_{4}-R_{3}={\begin{bmatrix}1&1&-1&0\\0&0&1&1\\0&0&b+6&0\\b-2&0&0&-2\end{bmatrix}}\,

If b=2,|A|=1\cdot 0\cdot 8\cdot (-2)=0\,,Therefore,the rank of A=3.

If b=-6\, number of non-zero rows is 3,rank of A=3.

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