Solve the equation $\left|\begin{array}{ccc}x+a & x & x \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0$,where $a \neq 0$.

If $a_i^2 + b_i^2 + c_i^2 = 1$ for $i = 1, 2, 3$ and $a_ia_j + b_ib_j + c_ic_j = 0$ for $i \ne j$ where $i, j = 1, 2, 3$,then the value of the determinant $\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|$ is:

The value of $\left|\begin{array}{lll}x & p & q \\ p & x & q \\ p & q & x\end{array}\right|$ is

If $(x, y, z) \ne (0, 0, 0)$ and $(i + j + 3k)x + (3i - 3j + k)y + (-4i + 5j)z = \lambda (xi + yj + zk),$ then the value of $\lambda$ will be

Evaluate the determinant: $\left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right|$

$\left| {\begin{array}{ccc} b + c & a - b & a \\ c + a & b - c & b \\ a + b & c - a & c \end{array}} \right| = $