Question

Here is a matrix `((1, t, t^2), (0, 1, 2t), (t, 0, 2))` ,Does there exist a value of `t` for which this matrix fails to have an inverse? Explain. Can someone help me solve this?

I started with the fact that if `det(A)=0` then `A^(-1)` does not exist .
With my calculation `t=root(3)(-2)`. Please correct me if I'm wrong .

Answer 1

`t = root(3)(-2)" "` or `" "t = omega root(3)(-2)" "` or `" "t = omega^2 root(3)(-2)`

where `omega = -1/2+sqrt(3)/2i` is the primitive complex cube root of `1`.

Explanation:

`abs((1, t, t^2), (0, 1, 2t), (t, 0, 2)) = 1abs((1,2t),(0,2))+t abs((2t,0),(2,t)) = t^2 abs((0,1),(t,0))`

`color(white)(abs((1, t, t^2), (0, 1, 2t), (t, 0, 2))) = 2+2t^3-t^3 = t^3+2`

So `abs((1, t, t^2), (0, 1, 2t), (t, 0, 2)) = 0` if and only if `t^3 = -2`

That is, if:

`t = root(3)(-2)" "` or `" "t = omega root(3)(-2)" "` or `" "t = omega^2 root(3)(-2)`

where `omega = -1/2+sqrt(3)/2i` is the primitive complex cube root of `1`.