Answer
the third root of $B$ is $A=\left[\begin{array}{ccc}{2} &{0}&{0} \\{0}& {-1} &{0}\\{0}&{0}&{3} \end{array}\right]$.
Work Step by Step
Let the matrix $A$ be given by
$$A=\left[\begin{array}{ccc}{2} &{0}&{0} \\{0}& {-1} &{0}\\{0}&{0}&{3} \end{array}\right].$$
Since $$A^3=\left[\begin{array}{ccc}{8} &{0}&{0} \\{0}& {-1} &{0}\\{0}&{0}&{27} \end{array}\right] =B,$$
then the third root of $B$ is $A=\left[\begin{array}{ccc}{2} &{0}&{0} \\{0}& {-1} &{0}\\{0}&{0}&{3} \end{array}\right]$.
