First slide

Introduction to Determinants

Question

If $x = - 9$ is a root of $|\begin{array}{lll} x 3 7 \\ 2 x 2 \\ 7 6 x \end{array}| = 0$ then the other two roots are

Moderate

A

3, 7
B

2, 7
C

3, 6
D

2, 6

Solution

Applying , $R_{1} \to R_{1} + R_{2} + R_{3}$ we get
$Δ = |\begin{array}{ccc} x + 9 & x + 9 & x + 9 \\ 2 & x & 2 \\ \bar{7} & 6 & x \end{array}|$
Taking x + 9 common from R1 and applying $C_{2} \to C_{2} - C_{1}, C_{3} \to {C_{3}}_{} - C_{1}$ we get
$Δ = (x + 9) |\begin{array}{ccc} 1 & 0 & 0 \\ 2 & x - 2 & 0 \\ 7 & - 1 & x - 7 \end{array}|$
$= (x + 9) (x - 2) (x - 7)$
Thus , $∆ = 0 \Rightarrow x = - 9, 2, 7$
$∴$ the other two roots are 2 and 7.

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