Question

The coefficient of $x^2$ in the expansion of the determinant $\left|\begin{array}{ccc}{x}^2& x^3+1& x^5+2\\ x^2+3& x^3+x& x^3+x^4\\ x+4& x^3+x^5& 2^3\end{array}\right|$ is

• A. -10
• B. -8
• C. -2
• D. -6
• E. 8

Answer 1

Answer: (A)

For the coefficient of $x^2$, on expanding along $R_1$, we get
$\Delta =x^2\left[8x^2+8x-x^6-2x^7-x^8\right]-\left(x^3+1\right)$
$\left[8x^3+24-x^4-x^5-4x^3-4x^4\right]+\left(x^5+2\right)$
$\left[x^6+x^7+3x^3+3x^4-x^3-x^2-4x^2-4x\right]$
$=8x^4+8x^3-x^8-2x^9-x^{10}-8x^6$
$-24x^3+x^7+x^8+4x^6+4x^7-8x^3-24$
$+x^4+x^5+4x^3+4x^4+x^{11}+x^{12}$
$+3x^8+3x^9-x^8-x^7-4x^7-4x^6+2x^6$
$+2x^7+6x^3+6x^4-2x^3-2x^2-8x^2-8x$
Coefficient of $x^2=-2-8=-10$