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Q.
If $a, b, c$ are real then the value of determinant $\begin{vmatrix}a^2+1 & a b & a c \\ a b & b^2+1 & b c \\ a c & b c & c^2+1\end{vmatrix}=1$ if
Determinants
Solution:
Multiply $R_1$ by a, $R_2$ by b & $R_3$ by $c \&$ divide the determinant by abc. Now take $a, b$ & c common from $c _1, c _2 \& c _3$. Now use $C _1 \rightarrow C _1+ C _2+ C _3$ to get ]
$\left(a^2+b^2+c^2+1\right)\begin{vmatrix}1 & 1 & 1 \\ b^2 & b^2+1 & b^2 \\ c^2 & c^2 & c^2+1\end{vmatrix}=1$.
Now use $c_1 \rightarrow c_1-c_2 \& c_2 \rightarrow c_2-c_3$
we get $1+a^2+b^2+c^2=1 \Rightarrow a=b=c=0 \Rightarrow$ (D)