Question

Factors of the determinant \(\left| {\begin{array}{*{20}{c}} a&{b + c}&{{a^2}}\\ b&{c + a}&{{b^2}}\\ c&{a + b}&{{c^2}} \end{array}} \right|\)
1. (a - b), (b - c), (c - a), (a + b + c)
2. (a + b), (b + c), (c + a), (a + b + c)
3. (a + b), (b - c), (c + a), (a + b + c)
4. (a² + b²), (b² + c²), (c² + a²)

Answer 1

Correct Answer - Option 1 : (a - b), (b - c), (c - a), (a + b + c)

\(\left| {\begin{array}{*{20}{c}} a&{b + c}&{{a^2}}\\ b&{c + a}&{{b^2}}\\ c&{a + b}&{{c^2}} \end{array}} \right|\)

C₁ → C₁ + C₂

\(=\left| {\begin{array}{*{20}{c}} {a + b + c}&{b + c}&{{a^2}}\\ {a + b + c}&{c + a}&{{b^2}}\\ {a + b + c}&{a + b}&{{c^2}} \end{array}} \right|\)

\(\Rightarrow {\left( {a + b + c} \right)\left| {\begin{array}{*{20}{c}} 1&{b + c}&{{a^2}}\\ 1&{c + a}&{{b^2}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)

∴ (a + b + c) is one of the factor.

\(\Rightarrow \left( {a + b + c} \right){\left| {\begin{array}{*{20}{c}} 1&{b + c}&{{a^2}}\\ 1&{c + a}&{{b^2}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)

R₁ → R₁ - R₂ and R₂ → R₂ - R₃

\(\Rightarrow\left( {a + b + c} \right) {\left| {\begin{array}{*{20}{c}} 0&{b - a}&{{a^2 - b^2}}\\ 0&{c - b}&{{b^2-c^2}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)

R1 → R1 - R2 

\(\Rightarrow\left( {a + b + c} \right){\left| {\begin{array}{*{20}{c}} 0&{c - a}&{{a^2 - c^2}}\\ 0&{c - b}&{{b^2-c^2}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)

\(\Rightarrow\left( {a + b + c} \right) {\left( {c-a})( {b-c} \right)\left| {\begin{array}{*{20}{c}} 0&{1}&{{-(a + c)}}\\ 0&{-1}&{{b+c}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|}\)

R₁ → R ₁ + R₂

\(\Rightarrow \left( {a + b + c} \right){\left( {c-a})( {b-c} \right)\left| {\begin{array}{*{20}{c}} 0&{0}&{{b-a}}\\ 0&{-1}&{{b+c}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)

\(\Rightarrow \left( {a + b + c} \right){\left( {c-a})( {b-c})(a-b \right)\left| {\begin{array}{*{20}{c}} 0&{0}&{{-1}}\\ 0&{1}&{{b+c}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)

∴ (a - b), (b - c), (c - a), (a + b + c) are the factors.