Question

If \(\rm \begin{vmatrix} a & -b & a - b - c\\ -a & b & -a + b - c\\ -a & -b & -a - b + c \end{vmatrix} - kabc = 0\) (a ≠ 0, b ≠ 0, c ≠ 0)

then what is the value of k?

1. -4
2. -2
3. 2
4. 4

Answer 1

Correct Answer - Option 1 : -4

Formula used:

\(\rm A = \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix}\)

det(A) = a(ei - hf) - b(di - gf) + c(dh - eg)

Calculation:

\(\begin{vmatrix} a & -b & a - b - c\\ -a & b & -a + b - c\\ -a & -b & -a - b + c \end{vmatrix} = kabc \)

R₂ → R₂ + R₁

\(\begin{vmatrix} a & -b & a - b - c\\ 0 & 0 & - 2c\\ -a & -b & -a - b + c \end{vmatrix} = kabc \)

⇒ a[-(-2c)(-b)] - (-b)[-(-2c)(-a)]

⇒ - 2abc - 2abc + (a - b - c ) × 0 = kabc

⇒ - 4abc = kabc

⇒ k = - 4

∴ The value of k is - 4