Consider the system of equations ax+y+bz=0, bx+y+az=0 and ax+by+abz=0 where a,b,∈0,1,2,3,4. The number of ordered pair (a, b) for which the system has non-trivial solution is___

a1bb1aabab=0⇒(a−b)a−b2=0 ⇒a=b (or) a=b2↓(0,0), (1,1), (2,2), (3,3),(4,4): (4,2)No. of ordered pairs =6

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Consider the system of equations $a x + y + b z = 0,$ $b x + y + a z = 0$ and $a x + b y + a b z = 0$ where $a, b, \in \{0, 1, 2, 3, 4\}$ . The number of ordered pair (a, b) for which the system has non-trivial solution is___

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answer is 6.

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$|\begin{array}{ccc} a & 1 & b \\ b & 1 & a \\ a & b & a b \end{array}| = 0 \Rightarrow (a - b) (a - b^{2}) = 0$

$\begin{array}{l} \Rightarrow a = b (o r) a = \underset{↓}{b^{2}} \\ (0, 0), (1, 1), (2, 2), (3, 3), (4, 4) : (4, 2) \end{array}$

No. of ordered pairs =6

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