Question

The value of $\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ba& b^2+1& bc\\ ca& cb& c^2+1\end{array}\right|$ is

• A. ${\left(a+b+c\right)}^2$
• B. $a^2+b^2+c^2$
• C. $a^2+b^2+c^2+1$
• D. $1$

Answer 1

The correct option is C

$a^2+b^2+c^2+1$

Explanation for correct option

$$\begin{gathered} \left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ba& b^2+1& bc\\ ca& cb& c^2+1\end{array}\right| \\ =\frac{1}{abc}\left|\begin{array}{ccc}a\left(a^2+1\right)& a^{2}b& a^{2}c\\ b^{2}a& b\left(b^2+1\right)& b^{2}c\\ c^{2}a& c^{2}b& c\left(c^2+1\right)\end{array}\right|;\left[R_1\to a{R}_1,R_2\to b{R}_2,R_3\to c{R}_3\right] \\ =\left|\begin{array}{ccc}\left(a^2+1\right)& a^2& a^2\\ b^2& \left(b^2+1\right)& b^2\\ c^2& c^2& \left(c^2+1\right)\end{array}\right|;\left[C_1\to \frac{1}{a}{C}_1,C_2\to \frac{1}{b}{C}_2,C_3\to \frac{1}{c}{C}_3\right] \\ =\left|\begin{array}{ccc}\left(a^2+b^2+c^2+1\right)& \left(a^2+b^2+c^2+1\right)& \left(a^2+b^2+c^2+1\right)\\ b^2& \left(b^2+1\right)& b^2\\ c^2& c^2& \left(c^2+1\right)\end{array}\right|;\left(R_1=R_1+R_2+R_3\right) \\ =\left(a^2+b^2+c^2+1\right)\left|\begin{array}{ccc}1& 1& 1\\ b^2& \left(b^2+1\right)& b^2\\ c^2& c^2& \left(c^2+1\right)\end{array}\right| \\ =\left(a^2+b^2+c^2+1\right)\left|\begin{array}{ccc}1& 0& 0\\ b^2& 1& 0\\ c^2& 0& 1\end{array}\right|;\left(C_2\to C_2-C_1,C_3\to C_3-C_1\right) \end{gathered}$$

$=\left(a^2+b^2+c^2+1\right)\left[1\left(1\times 1-0\right)\right]$ [Expanding along $R_1$ we get]

$=\left(a^2+b^2+c^2+1\right)$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{C}\right)$