Determine the values of a and b for which the system X+2y+3z =6 X+3y+5z =9 2x+5y+az =b Has 1) no solution 2) unique solution 3) infinite solution

[tex]\sf \: \left[\begin{array}{ccc}1&2&3\\1&3&5\\2&5& a\end{array}\right] \: \left[\begin{array}{c}x\\y\\z\end{array}\right] \: = \: \left[\begin{array}{c}6\\9\\b\end{array}\right] \\ [/tex]

where,

[tex]\sf \: A = \left[\begin{array}{ccc}1&2&3\\1&3&5\\2&5& a\end{array}\right] \\ [/tex]

[tex]\sf \: B = \left[\begin{array}{c}6\\9\\b\end{array}\right] \\ [/tex]

[tex]\sf \: X = \left[\begin{array}{c}x\\y\\z\end{array}\right]\\ [/tex]

Now, Consider

[tex]\sf \: \rho[A:B ]\\ [/tex]

[tex]\qquad\sf \: = \: \left[\begin{array}{ccc}1&2&3\: \: :&6\\1&3&5\: \: :&9\\2&5& a\: \: :&b\end{array}\right]\\ [/tex]

[tex]\qquad\qquad\sf \: OP\:R_2\:\to\:R_2 - R_1 \\ [/tex]

[tex]\qquad\sf \: = \: \left[\begin{array}{ccc}1&2&3\: \: :&6\\0&1&2\: \: :&3\\2&5& a\: \: :&b\end{array}\right] \\ [/tex]

[tex]\qquad\qquad\sf \: OP\:R_3\:\to\:R_3 - 2R_1\\ [/tex]

[tex]\qquad\sf \: = \: \left[\begin{array}{ccc}1&2&3\: \: :&6\\0&1&2\: \: :&3\\0&1& a - 6\: \: :&b - 12\end{array}\right] \\ [/tex]

[tex]\qquad\qquad\sf \: OP\:R_3\:\to\:R_3 - R_2\\ [/tex]

[tex]\qquad\sf \: = \: \left[\begin{array}{ccc}1&2&3\: \: :&6\\0&1&2\: \: :&3\\0&0& a - 8\: \: :&b - 15\end{array}\right] \\ [/tex]

Now, for system of linear equations has no solution, we have

[tex]\sf \: \rho(A) \ne \rho[ A:B] \\ [/tex]

So, it means

[tex]\sf \: a - 8 = 0 \: \: and \: \: b - 15 \ne0\\ [/tex]

[tex]\sf\implies \bf \: a = 8 \: \: and \: \: b \ne 15\\ [/tex]

Now, we know, for system of linear equations has unique solution.

[tex]\sf \: \rho(A) = \rho[ A:B] = number \: of \: variables \\ [/tex]

So, using this result, we get

[tex]\sf \: \rho(A) = \rho[ A:B] = 3\\ [/tex]

So, it means

[tex]\sf \: a - 8 \ne 0 \: \: and \: \: b \in R\\ [/tex]

[tex]\sf\implies \bf \: a = 8 \: \: and \: \: b \in R\\ [/tex]

Now, we know, for system of linear equations has infinitely many solutions.

[tex]\sf \: \rho(A) = \rho[ A:B] < number \: of \: variables \\ [/tex]

So, using this result, we get

[tex]\sf \: \rho(A) = \rho[ A:B] < 3\\ [/tex]

So, it means

[tex]\sf \: a - 8 = 0 \: \: and \: \: b - 15 = 0\\ [/tex]

[tex]\sf\implies \bf \: a = 8 \: \: and \: \: b = 15\\ [/tex]