91Ó°ÊÓ

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

Find the values of \(\lambda\) and \(\mu\) so that the system of equation \(x+y+z=6\) \(x+2 y+3 z=10\) \(x+2 y+\lambda z=\mu\) has i. Unique solution \\{Ans. \(\lambda \neq 3, \mu \in R\\}\) ii. Infinite solutions \\{Ans. \(\lambda=3, \mu=10\\}\) iii. No solution \\{Ans. \(\lambda=3, \mu \neq 10\\}\)

If \(\left[\begin{array}{ll}x & 0 \\ 1 & y\end{array}\right]+\left[\begin{array}{cc}-2 & 1 \\ 3 & 4\end{array}\right]=\left[\begin{array}{ll}3 & 5 \\ 6 & 3\end{array}\right]-\left[\begin{array}{ll}2 & 4 \\ 2 & 1\end{array}\right]\), then find \(x, y .\)

Given \(A=\left[\begin{array}{ccc}1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1\end{array}\right]\) and \(B=\left[\begin{array}{ccc}3 & -1 & 2 \\ 4 & 2 & 5 \\\ 2 & 0 & 3\end{array}\right]\). Find the matrix \(C\) such that \(A+2 C=B\).

Show that if \(A\) is an orthogonal matrix, then \(A^{T}\) is also orthogonal.

Find all idempotent diagonal matrices of order 3 .