Question

evaluate the following determinants
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Answer 1

Answer:

Answer (a)

$\left|\begin{array}{ccc}1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25\end{array}\right|$

=1(225-256)-4(100-144)+9(64-81)

=-21+176-135

= -8

Answer (b)

Expanding the determinant along first row, we get

$\[ \begin{array}{l} \rm \left|\begin{array}{ll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|=a\left|\begin{array}{ll} b & f \\ f & c \end{array}\right|-h\left|\begin{array}{ll} h & f \\ g & c \end{array}\right|+g\left|\begin{array}{ll} h & b \\ g & f \end{array}\right| \\ \\ \rm=a\left(b c-f^{2}\right)-h(c h-f g)+g(h f-b g) \\ \\ \rm=a b c-a f^{2}-c h^{2}+f g h+f g h-b g^{2} \\ \\ \rm=a b c+2 f g h-a f^{2}-b g^{2}-c h^{2} . \end{array} \]$