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3 and 4 .Determinants and Matrices
medium
If $-9 $ is a root of the equation $\left| {\,\begin{array}{*{20}{c}}x&3&7\\2&x&2\\7&6&x\end{array}\,} \right| = 0$ then the other two roots are
A
$2, 7$
B
$-2, 7$
C
$2, -7$
D
$-2, -7$
(IIT-1983)
Solution
(a) $\left| {\,\begin{array}{*{20}{c}}x&3&7\\2&x&2\\7&6&x\end{array}\,} \right|\, = 0$
$ \Rightarrow $ $(x + 9)\,\left| {\,\begin{array}{*{20}{c}}1&1&1\\2&x&2\\7&6&x\end{array}\,} \right| = 0$,
by ${R_1} \to {R_1} + {R_2} + {R_3}$
$ \Rightarrow $ $(x + 9)\,\{ ({x^2} – 12) – (2x – 14) + (12 – 7x)\} = 0$
$ \Rightarrow $ $(x + 9)\,({x^2} – 9x + 14) = 0$
$ \Rightarrow (x + 9)(x – 2)\,(x – 7) = 0$
Hence the other two roots are $x = 2,\,7$.
Standard 12
Mathematics
