Question

What is the value of the determinant \(\left |\begin{array}{ccc} \rm x+2 & \rm x+3 & \rm x-1 \\ \rm x+6 &\rm x+8 &\rm x+4 \\ \rm x+9 &\rm x+11 &\rm x+7 \end{array}\right|\)
1. -x + 32
2. 32
3. 12
4. 16

Answer 1

Correct Answer - Option 3 : 12

Calculation:

Let Δ = \(\left |\begin{array}{ccc} \rm x+2 & \rm x+3 & \rm x-1 \\ \rm x+6 &\rm x+8 &\rm x+4 \\ \rm x+9 &\rm x+11 &\rm x+7 \end{array}\right|\)

By applying C₂ → C₂ - C₁, C₃ → C₃ - C₁

\(\begin{aligned} =\left|\begin{array}{ccc} \rm x+2 & 1 & -3 \\ \rm x+6 & 2 & -2 \\ \rm x+9 & 2 & -2 \end{array}\right| \end{aligned}\)

By applying C3 → C3 + C₂

\(\begin{aligned} =\left|\begin{array}{ccc} \rm x+2 & 1 & -2 \\ \rm x+6 & 2 & 0 \\ \rm x+9 & 2 & 0 \end{array}\right| \end{aligned}\)

Expanding along C₃, we get

= -2 [2(x + 6) - 2(x + 9)]

= -4 [ x + 6 - x - 9]

= 12

Hence, option (3) is correct.