LA2.1.13

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Show that \begin{vmatrix} a & a+b & a+b+c \\ 2a 3a+2b & 4a+3b+2c \\ 3a & 6a+3b & 10a+6b+3c \end{vmatrix}=a^3\,

Applying R_2-2R_1,R_3-3R_1\, to the given,we get

\begin{vmatrix} a & a+b & a+b+c \\ 2a 3a+2b & 4a+3b+2c \\ 3a & 6a+3b & 10a+6b+3c \end{vmatrix}=\begin{vmatrix} a & a+b & a+b+c \\ 0 & a & 2a+b \\ 0 & 3a & 7a+3b \end{vmatrix} \,

a[a(7a+3b)-3a(2a+b)]=a^3\,=RHS


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