Evaluate the following determinant:

Solution:

$(\mathrm{i}) \Delta=\left|\begin{array}{ccc}1 & 3 & 5 \\ 2 & 6 & 10 \\ 31 & 11 & 38\end{array}\right|$

$=1\left|\begin{array}{cc}6 & 10 \\ 11 & 38\end{array}\right|-3\left|\begin{array}{cc}2 & 10 \\ 31 & 38\end{array}\right|+5\left|\begin{array}{cc}2 & 6 \\ 31 & 11\end{array}\right|$

$=1(228-110)-3(76-310)+5(22-186)$

$=1(118)-3(-234)+5(-164)$

$=118+702-820$

$=0$

$($ ii $) \Delta=\left|\begin{array}{lll}67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26\end{array}\right|$

$=67(338-336)-19(1014-1134)+21(936-1053)$

$=67(2)-19(-120)+21(-117)$

$=134+2280-2457$

$=-43$

$($ iii $) \Delta=\left|\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right|$

$=a\left|\begin{array}{ll}b & f \\ f & c\end{array}\right|-h\left|\begin{array}{ll}h & f \\ g & c\end{array}\right|+g\left|\begin{array}{ll}h & b \\ g & f\end{array}\right|$

$=a\left(b c-f^{2}\right)-h(h c-f g)+g(h f-g b)$

$=a b c-a f^{2}-h^{2} c+f g h+f g h-g^{2} b$

$=a b c+2 f g h-a f^{2}-c h^{2}-b g^{2}$

$($ iv $) \Delta=\left|\begin{array}{ccc}1 & -3 & 2 \\ 4 & -1 & 2 \\ 3 & 5 & 2\end{array}\right|$

$=1\left|\begin{array}{cc}-1 & 2 \\ 5 & 2\end{array}\right|-(-3)\left|\begin{array}{cc}4 & 2 \\ 3 & 2\end{array}\right|+2\left|\begin{array}{cc}4 & -1 \\ 3 & 5\end{array}\right|$

$=1(-2-10)+3(8-6)+2(20+3)$

$=(-12)+6+46$

$=40$

$(\mathrm{v}) \Delta=\left|\begin{array}{ccc}1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25\end{array}\right|$

$=1\left|\begin{array}{cc}9 & 16 \\ 16 & 25\end{array}\right|-4\left|\begin{array}{ll}4 & 16 \\ 9 & 25\end{array}\right|+9\left|\begin{array}{cc}4 & 9 \\ 9 & 16\end{array}\right|$

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

$=1(-31)-4(-44)+9(-17)$

$=-31+176-153$

$=-8$

$(\mathrm{vi}) \Delta=\left|\begin{array}{ccc}6 & -3 & 2 \\ 2 & -1 & 2 \\ -10 & 5 & 2\end{array}\right|$

$=6(-2-10)-(-3)(4+20)+2(10-10)$

$=-72+72+0$

$=-72+72$

$=0$

$($ vii $) \Delta=\left|\begin{array}{cccc}1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9\end{array}\right|$

$=1\left|\begin{array}{ccc}9 & 27 & 1 \\ 27 & 1 & 3 \\ 1 & 3 & 9\end{array}\right|-3\left|\begin{array}{ccc}3 & 27 & 1 \\ 9 & 1 & 3 \\ 27 & 3 & 9\end{array}\right|+9\left|\begin{array}{ccc}3 & 9 & 1 \\ 9 & 27 & 3 \\ 27 & 1 & 9\end{array}\right|-27\left|\begin{array}{ccc}3 & 9 & 27 \\ 9 & 27 & 1 \\ 27 & 1 & 3\end{array}\right|$

$=1\{9(9-9)-27(243-3)+1(81-1)\}-3\{3(9-9)-27(81-81)+1(27-27)\}+9\{3(243-3)-9(81-81)+1(9-729)\}$

$-27\{(81-1)-9(27-27)+27(9-729)\}$

$=1\{0-6480+80\}-3\{0-0+0\}+9\{720-0-720\}-27\{80-0-19440\}$

$=-6400+522720$

$=516320$