The value of a for which the system of equati | vedclass

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Question

(a) The system will have a non-zero solution, if
$\Delta \equiv \left| {\,\begin{array}{*{20}{c}}{{a^3}}&{{{(a + 1)}^3}}&{{{(a + 2)}^3}}\a&a

Vedclass · Accepted Answer

$-1$

Vedclass · Answer

(a) The system will have a non-zero solution, if
$\Delta \equiv \left| {\,\begin{array}{*{20}{c}}{{a^3}}&{{{(a + 1)}^3}}&{{{(a + 2)}^3}}\a&{a + 1}&{a + 2}\1&1&1\end{array}\,} ight| = 0$

$ \Rightarrow \left| {\,\begin{array}{*{20}{c}}{{a^3}}&{3{a^2} + 3a + 1}&{3{{(a + 1)}^2} + 3(a + 1) + 1}\{{a^2}}&1&1\1&0&0\end{array}\,} ight| = 0$

by $\begin{array}{l}{C_2} 	o {C_2} - {C_1}\{C_3} 	o {C_3} - {C_2}\end{array}$

==> $3{a^2} + 3a + 1 - \{ 3{(a + 1)^2} + 3(a + 1) + 1\} $

(expanding along ${R_3}$)

==> $ - 6(a + 1) = 0 \Rightarrow a = - 1$.

The value of a for which the system of equations ${a^3}x + {(a + 1)^3}y + {(a + 2)^3}z = 0,$ $ax + (a + 1)y + (a + 2)z = 0,$ $x + y + z = 0,$ has a non zero solution is

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