Question

Suppose $x_1, x_2, x_3$ are real numbers such that $x_1 x_2 x_3 \neq 0$.
Let $\Delta=\begin{vmatrix} x_1+a_1 b_1 & a_1 b_2 & a_1 b_3 \\ a_2 b_1 & x_2+a_2 b_2 & a_2 b_3 \\ a_3 b_1 & a_3 b_2 & x_3+a_3 b_3 \end{vmatrix}$
Then $\frac{\Delta}{x_1 x_2 x_3}-1$ equals:

• A. $\frac{a_1 b_1}{x_1}+\frac{a_2 b_2}{x_2}+\frac{a_3 b_3}{x_3}$
• B. -1
• C. $\frac{a_1 a_2 a_3+b_1 b_2 b_3}{x_1 x_2 x_3}$
• D. 0

Answer 1

Answer: (A)

Using the sum property, write
$\Delta=x_1 \Delta_1+b_1 \Delta_2$
where
$\Delta_1 =\begin{vmatrix}1 & a_1 b_2 & a_1 b_3 \\0 & x_2+a_2 b_2 & a_2 b_3 \\0 & a_3 b_2 & x_3+a_3 b_3
\end{vmatrix}$
$ =\left(x_2+a_2 b_2\right)\left(x_3+a_3 b_3\right)-a_3 b_2 a_2 b_3$
$ =x_2 x_3+x_2 a_3 b_3+x_3 a_2 b_2$
$\Delta_2 =\begin{vmatrix}a_1 & a_1 b_2 & a_1 b_3 \\a_2 & x_2+a_2 b_2 & a_2 b_3 \\a_3 & a_3 b_2 & x_3+a_3 b_3
\end{vmatrix}$
Using $C_2 \rightarrow C_2-b_2 C_1, C_3 \rightarrow C_3-b_3 C_1$, we get
$\Delta_2=\begin{vmatrix}a_1 & 0 & 0 \\a_2 & x_2 & 0 \\a_3 & 0 & x_3
\end{vmatrix}=a_1 x_2 x_3$
Thus,
$\Delta =x_1\left[x_2 x_3+x_2 a_3 b_3+x_3 a_2 b_2\right]+a_1 b_1 x_2 x_3 $
$\Rightarrow \frac{\Delta}{x_1 x_2 x_3}-1 =\frac{a_1 b_1}{x_1}+\frac{a_2 b_2}{x_2}+\frac{a_3 b_3}{x_3}$