Introduction of Determinant

Given that:  A = `((1,-1,0),(2,3,4),(0,1,2))`  and B = `((2,2,-4),(-4,2,-4),(2,-1,5))`  , find AB. Using this result, solve the following system of equation: x – y = 3, 2x + 3y + 4z = 17 and y + 2z = 7. 

If the equations 2x + 3y + z = 0, 3x + y - 2z = 0 and ax + 2y - bz = 0 has non-trivial solution, then ____________.

A non-trivial solution of the system of equations x + λy + 2z = 0, 2x + λz = 0, 2λx - 2y + 3z = 0 is given by x : y : z = ____________.

Find the value of the determinant given below, without expanding it at any stage.

`|(βγ, 1, α(β + γ)),(γα, 1, β(γ + α)),(αβ, 1, γ(α + β))|`

If ω is non real cube root of unity, then `abs ((2, 2omega, -omega^2),(1,1,1),(1,-1,0))` is equal to ____________.

Find the area of the triangle with vertices P(4, 5), Q(4, -2), and R(-6, 2).

If `omega` is a non-real cube root of unity and n is not a multiple of 3, then `Delta = abs ((1, omega^n, omega^(2n)),(omega^(2n), 1, omega^n),(omega^n, omega^(2n), 1))` is equal to ____________.

Write the value of the determinant \[\begin{vmatrix}p & p + 1 \\ p - 1 & p\end{vmatrix}\]