Prove the following using properties of determinants : \[\begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix} = 2\left( a + b + c \right) {}^3\]

Let \[\bigtriangleup = \begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix}\] Applying \[C_1 \to C_1 + C_2 + C_3\] , we get: \[\bigtriangleup = \begin{vmatrix}2a + 2b + 2c & a & b \\ 2a + 2b + 2c & b + c + 2a & b \\ 2a + 2b + 2c & a & c + a + 2b\end{vmatrix}\] \[\Rightarrow \bigtriangleup = 2\left( a + b + c \right)\begin{vmatrix}1 & a & b \\ 1 & b + c + 2a & b \\ 1 & a & c + a + 2b\end{vmatrix}\] Now, applying \[R_2 \to R_2 - R_1 \text { and } R_3 \to R_3 - R_1\] , we get: \[\Rightarrow \bigtriangleup = 2\left( a + b + c \right)\begin{vmatrix}1 & a & b \\ 0 & b + c + a & 0 \\ 0 & 0 & c + a + b\end{vmatrix}\] \[\Rightarrow \bigtriangleup = 2 \left( a + b + c \right)^3\] &there4; \[\begin{vmatrix}a + b + 2c & a & b \\ c & b + c + 2a & b \\ c & a & c + a + 2b\end{vmatrix} = 2\left( a + b + c \right) {}^3\]

Prove the Following Using Properties of Determinants : ∣ ∣ ∣ ∣ a + B + 2 C a B C B + C + 2 a B C a C + a + 2 B ∣ ∣ ∣ ∣ = 2 ( a + B + C ) 3 - Mathematics

Question

Prove the following using properties of determinants :

$| \begin{matrix} a + b + 2 c & a & b \\ c & b + c + 2 a & b \\ c & a & c + a + 2 b \end{matrix} | = 2 (a + b + c)^{3}$

Solution

Let $△ = | \begin{matrix} a + b + 2 c & a & b \\ c & b + c + 2 a & b \\ c & a & c + a + 2 b \end{matrix} |$

Applying $C_{1} \to C_{1} + C_{2} + C_{3}$ , we get:

$△ = | \begin{matrix} 2 a + 2 b + 2 c & a & b \\ 2 a + 2 b + 2 c & b + c + 2 a & b \\ 2 a + 2 b + 2 c & a & c + a + 2 b \end{matrix} |$

$\Rightarrow △ = 2 (a + b + c) | \begin{matrix} 1 & a & b \\ 1 & b + c + 2 a & b \\ 1 & a & c + a + 2 b \end{matrix} |$

Now, applying

$R_{2} \to R_{2} - R_{1} and R_{3} \to R_{3} - R_{1}$ , we get:

$\Rightarrow △ = 2 (a + b + c) | \begin{matrix} 1 & a & b \\ 0 & b + c + a & 0 \\ 0 & 0 & c + a + b \end{matrix} |$

$\Rightarrow △ = 2 {(a + b + c)}^{3}$

∴ $| \begin{matrix} a + b + 2 c & a & b \\ c & b + c + 2 a & b \\ c & a & c + a + 2 b \end{matrix} | = 2 (a + b + c)^{3}$

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Properties of Determinants

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Q 19 Q 18 Q 20

2013-2014 (March) Delhi Set 2

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2013-2014 (March) Delhi Set 2 (with solutions)

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