Chapter 10

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} 1 & 2 & 3 \\ 3 & 5 & 7 \\ 8 & 14 & 20 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} 43 & 1 & 6 \\ 35 & 7 & 4 \\ 17 & 3 & 2 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} 11 & 12 & 13 \\ 12 & 13 & 14 \\ 13 & 14 & 15 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} 38 & 7 & 63 \\ 16 & 3 & 29 \\ 27 & 5 & 46 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} 265 & 240 & 219 \\ 240 & 225 & 198 \\ 219 & 198 & 181 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} 18 & 40 & 89 \\ 40 & 89 & 198 \\ 89 & 198 & 440 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} \sqrt{13}+\sqrt{3} & 2 \sqrt{5} & \sqrt{5} \\ \sqrt{15}+\sqrt{26} & 5 & \sqrt{10} \\ 3+\sqrt{65} & \sqrt{15} & 5 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{cccc} 21 & 17 & 7 & 10 \\ 24 & 22 & 6 & 10 \\ 6 & 8 & 2 & 3 \\ 5 & 7 & 1 & 2 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{cccc} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 6 & 10 & 15 \\ 1 & 4 & 10 & 20 & 35 \\ 1 & 5 & 15 & 35 & 69 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} a-b & b-c & c-a \\ x-y & y-z & z-x \\ p-q & q-r & r-p \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} \frac{1}{a} & a^{2} & b c \\ \frac{1}{b} & b^{2} & c a \\ \frac{1}{c} & c^{2} & a b \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} \sin ^{2} x & \cos ^{2} x & 1 \\ \cos ^{2} x & \sin ^{2} x & 1 \\ -10 & 12 & 2 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} 1 & \log _{x} y & \log _{x} z \\ \log _{y} x & 1 & \log _{y} z \\ \log _{z} x & \log _{z} y & 1 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} b^{2}-a b & b-c & b c-a c \\ a b-a^{2} & a-b & b^{2}-a b \\ b c-a c & c-a & a b-a^{2} \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} 1 & a & a^{2}-b c \\ 1 & b & b^{2}-c a \\ 1 & c & c^{2}-a b \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} 0 & b & -c \\ -b & 0 & a \\ c & -a & 0 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{ccc} 0 & a-b & a-c \\ b-a & 0 & b-c \\ c-a & c-b & 0 \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{lll} 1 & b c & b c(b+c) \\ 1 & c a & c a(c+a) \\ 1 & a b & a b(a+b) \end{array}\right| $$

EVALUATING DETERMINANTS. $$ \left|\begin{array}{llll} 1 & a & a^{2} & a^{3}+b c d \\ 1 & b & b^{2} & b^{3}+c d a \\ 1 & c & c^{2} & c^{3}+a b d \\ 1 & d & d^{2} & d^{3}+a b c \end{array}\right| $$

If \(D_{r}=\left|\begin{array}{ccc}r & x & \frac{n(n+1)}{2} \\ 2 r-1 & y & n^{2} \\ 3 r-2 & z & \frac{n(3 n-1)}{2}\end{array}\right|\) then prove that \(\sum_{r=1}^{n} D_{r}=0 .\)

Let \(\Delta_{a}=\left|\begin{array}{ccc}a-1 & n & 6 \\ (a-1)^{2} & 2 n^{2} & 4 n-2 \\ (a-1)^{3} & 3 n^{2} & 3 n^{2}-3 n\end{array}\right|\) show that \(\sum_{a=1}^{n} \Delta_{a}=c\), a constant.

Evaluate \(\sum_{n=1}^{N} U_{n}\) if \(U_{n}=\left|\begin{array}{ccc}n & 1 & 5 \\\ n^{2} & 2 N+1 & 2 N+1 \\ n^{3} & 3 N^{2} & 3 N\end{array}\right| .\\{\)

Express \(\left|\begin{array}{ccc}1 & 2 & -3 \\ 2 & 1 & 1 \\ 2 & 3 & 1\end{array}\right|^{2}\) in determinant form and find its value also.

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{array}\right|=x y $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{array}\right|=\left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{array}\right|=(a-b)(b-c)(c-a) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} k a & k^{2}+a^{2} & 1 \\ k b & k^{2}+b^{2} & 1 \\ k c & k^{2}+c^{2} & 1 \end{array}\right|=k(a-b)(b-c)(c-a) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \begin{array}{lll} 1 & (m+n-l-p)^{2} & (m+n-l-p)^{4} \\ 1 & (n+l-m-p)^{2} & (n+l-m-p)^{4} \\ 1 & (l+m-n-p)^{2} & (l+m-n-p)^{4} \end{array} \mid=64(l-m)(l-n)(l-p)(m-n)(m-p)(n-p) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} a & b+c & a^{2} \\ b & c+a & b^{2} \\ c & a+b & c^{2} \end{array}\right|=-(a+b+c)(a-b)(b-c)(c-a) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{array}\right|=\left|\begin{array}{lll} 1 & b c & b+c \\ 1 & c a & c+a \\ 1 & a b & a+b \end{array}\right| $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ b+c & c+a & a+b \\ b^{2}+c^{2} & c^{2}+a^{2} & a^{2}+b^{2} \end{array}\right|=(b-c)(c-a)(a-b) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3} \end{array}\right|=(a-b)(b-c)(c-a)(a+b+c) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} -2 a & a+b & c+a \\ b+a & -2 b & b+c \\ c+a & c+b & -2 c \end{array}\right|=4(b+c)(c+a)(a+b) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{array}\right|=\left|\begin{array}{lll} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{array}\right| $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} x & y & z \\ -x & y & z \\ -x & -y & z \end{array}\right|=4 x y z $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{lll} b+c & c+a & a+b \\ a+b & b+c & c+a \\ c+a & a+b & b+c \end{array}\right|=2\left|\begin{array}{lll} a & b & c \\ c & a & b \\ b & c & a \end{array}\right| $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} 2 a b & a^{2} & b^{2} \\ a^{2} & b^{2} & 2 a b \\ b^{2} & 2 a b & a^{2} \end{array}\right|=-\left(a^{3}+b^{3}\right)^{2} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{array}\right|=4 x y z $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} y+z & x & x \\ y & z+x & y \\ z & z & x+y \end{array}\right|=4 x y z $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} \frac{a^{2}+b^{2}}{c} & c & c \\ a & \frac{b^{2}+c^{2}}{a} & a \\ b & b & \frac{c^{2}+a^{2}}{b} \end{array}\right|=4 a b c $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a & b & a x+b y \\ b & c & b x+c y \\ a x+b y & b x+c y & 0 \end{array}\right|=\left(b^{2}-a c\right)\left(a x^{2}+2 b x y+c y^{2}\right) $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a^{2} & b c & a c+c^{2} \\ a^{2}+a b & b^{2} & a c \\ a b & b^{2}+b c & c^{2} \end{array}\right|=4 a^{2} b^{2} c^{2} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} b^{2}+c^{2} & a b & a c \\ a b & c^{2}+a^{2} & b c \\ c a & c b & a^{2}+b^{2} \end{array}\right|=\left|\begin{array}{ccc} b^{2}+c^{2} & a^{2} & a^{2} \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right|=4 a^{2} b^{2} c^{2} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b \end{array}\right|=a^{3}+b^{3}+c^{3}-3 a b c $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|=(a+b+c)^{3} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{ccc} a+b+2 c & a & b \\ c & b+c+2 a & b \\ c & a & c+a+2 b \end{array}\right|=2(a+b+c)^{3} $$

PROVING IDENTITIES BY DETERMINANTS. $$ \left|\begin{array}{cccc} 1+a_{1} & a_{2} & a_{3} & a_{4} \\ a_{1} & 1+a_{2} & a_{3} & a_{4} \\ a_{1} & a_{2} & 1+a_{3} & a_{4} \\ a_{1} & a_{2} & a_{3} & 1+a_{4} \end{array}\right|=1+a_{1}+a_{2}+a_{3}+a_{4} $$