| The system of linear equations. |
| \[x+y+z=2\] |
| \[2x+3y+2z=5\] |
| \[2x+3y+({{a}^{2}}-1)z=a+1\] |
A) has infinitely many solutions for \[a=4\]
B) is inconsistent when \[\left| a \right|=\sqrt{3}\]
C) is inconsistent when \[a=4\]
D) has a unique solution for \[\left| a \right|=\sqrt{3}\]
Correct Answer: B
Solution :
| Augmented matrix [A : B] \[=\left[ \begin{matrix} 1 & 1 & 1 & 2 \\ 2 & 3 & 2 & 5 \\ 2 & 2 & {{a}^{2}}-1 & a+1 \\ \end{matrix} \right]\] |
| \[\text{By}\,{{\text{R}}_{2}}\,\to \,{{\text{R}}_{2}}\,-\,2{{\text{R}}_{1}}\,\text{and}\,{{\text{R}}_{3}}\,\to \,{{\text{R}}_{3}}\,-\,{{\text{R}}_{2}}\]\[\tilde{\ }\,\left[ \begin{matrix} 1 & 1 & 1 & 2 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & {{a}^{2}}-3 & a-4 \\ \end{matrix} \right]\] |
| For a \[a\,=\,\pm \sqrt{3}\]clearly |
| Rank of \[[A:B]=3\]and rank of\[A=2\] |
| Which are unequal \[\Rightarrow \]there exist no solution if \[\left| a \right|\,=\,\sqrt{3}.\] |
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