91Ó°ÊÓ

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