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Evaluate $$D=\begin{vmatrix} -2a &a+b &a+c \\ b+a& -2b &b+c \\ c+a&c+b & -2c \end{vmatrix}$$

My try:

Applying $R_1 \to R_1+R_2$ we get

$$D=\begin{vmatrix} b-a&a-b &a+b+2c \\ b+a& -2b &b+c \\ c+a&c+b & -2c \end{vmatrix}$$

Now apply $$C_1 \to C_1+C_2$$

$$D=\begin{vmatrix} 0&a-b &a+b+2c \\ a-b& -2b &b+c \\ 2c+a+b&c+b & -2c \end{vmatrix}$$

Now apply $C_2 \to C_2 +C_3$

$$D= \begin{vmatrix} 0&2a+2c &a+b+2c \\ a-b& c-b &b+c \\ 2c+a+b&b-c & -2c \end{vmatrix}$$

Now use $R_3 \to R_3+R_2$

$$D= \begin{vmatrix} 0&2a+2c &a+b+2c \\ a-b& c-b &b+c \\ 2c+2a&0 & b-c \end{vmatrix}$$

any way to proceed here using elementary operations?

StubbornAtom
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Ekaveera Gouribhatla
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1 Answers1

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Let $p(a,b,c) $ be the determinant. Note that each term of $p$ has degree 3 (sum of degrees of $a,b,c$).

Note that $p(a,-a,c) = 0$, hence $a+b$ divides $p$.

Similarly we see that $a+c, b+c$ divide $p$.

Hence $p$ has the form $p(a,b,c) = k (a+b)(b+c)(a+c)$ for some constant $k$.

Compute the determinant for $a=b=c={1\over 2}$ to get $k = 4$.

copper.hat
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