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  • Using the properties of determinants in evaluate: a-b-c,2a,2a

Using the properties of determinants in evaluate:
\left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|

Answers (1)

Let\: \: A=\left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|

\\\begin{aligned} &\text { Apply } R_{1} \rightarrow R_{1}+R_{2}+R_{y}\\ &=\left|\begin{array}{ccc} \mathrm{a}-\mathrm{b}-\mathrm{c}+2 \mathrm{~b}+2 \mathrm{c} & 2 \mathrm{a}+\mathrm{b}-\mathrm{c}-\mathrm{a}+2 \mathrm{c} & 2 \mathrm{a}+2 \mathrm{~b}+\mathrm{c}-\mathrm{a}-\mathrm{b} \\ 2 \mathrm{~b} & \mathrm{~b}-\mathrm{c}-\mathrm{a} & 2 \mathrm{~b} \\ 2 \mathrm{c} & 2 \mathrm{c} & \mathrm{c}-\mathrm{a}-\mathrm{b} \end{array}\right|\\ &=\left|\begin{array}{ccc} \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{a}+\mathrm{b}+\mathrm{c} & \mathrm{a}+\mathrm{b}+\mathrm{c} \\ 2 \mathrm{~b} & \mathrm{~b}-\mathrm{c}-\mathrm{a} & 2 \mathrm{~b} \\ 2 \mathrm{c} & 2 \mathrm{c} & \mathrm{c}-\mathrm{a}-\mathrm{b} \end{array}\right|\\ &\text { Take }(a+b+c) \text { common form the first row }\\ &=(a+b+c)\left|\begin{array}{ccc} 1 & 1 & 1 \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|\\ &\text { Apply } C_{2} \rightarrow C_{2}-C_{1} \end{aligned}

\begin{array}{l} =(a+b+c)\left|\begin{array}{ccc} 1 & 1-1 & 1 \\ 2 b & b-c-a-2 b & 2 b \\ 2 c & 2 c-2 c & c-a-b \end{array}\right| \\ =(a+b+c)\left|\begin{array}{ccc} 1 & 0 & 1 \\ 2 b & -b-c-a & 2 b \\ 2 c & 0 & c-a-b \end{array}\right| \end{array}

Now apply \mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-\mathrm{C}_{1}

\begin{array}{l} =(a+b+c)\left|\begin{array}{ccc} 1 & 0 & 1-1 \\ 2 b & -(a+b+c) & 2 b-2 b \\ 2 c & 0 & c-a-b-2 c \end{array}\right| \\ =(a+b+c)\left|\begin{array}{ccc} 1 & 0 & 0 \\ 2 b & -(a+b+c) & 0 \\ 2 c & 0 & -(a+b+c) \end{array}\right| \end{array}

Expand the determinant along Row 1

\begin{align*} &= (a + b + c)[1 \times \{ -(a + b + c) \times \{ -(a + b + c) \} - 0 \}] \\ &= (a + b + c)[(a + b + c)^2] \\ &= (a + b + c)^3 \end{align*}

 

 

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