The matrix \[{\text{P}} = \left[ {\begin{array}{*{20}{c}} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}} \right]\] rotates a vector about the axis \[\left[ {\begin{array}{*{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right]\] by angle of A. 30° B. 60° C. 90° D. 120°

The correct answer is $\boxed{\text{A}}$.

The matrix $\text{P}$ is a rotation matrix. It rotates a vector about the axis $\left[ {\begin{array}{*{20}{c}} 1 \ 1 \ 1 \end{array}} \right]$ by an angle of $90^\circ$.

To see this, let $v$ be a vector. Then, the vector $\text{P}v$ is given by

$$\text{P}v = \left[ {\begin{array}{{20}{c}} 0&0&1 \ 1&0&0 \ 0&1&0 \end{array}} \right] \left[ {\begin{array}{{20}{c}} v_x \ v_y \ v_z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} v_y \ v_z \ -v_x \end{array}} \right].$$

In other words, $\text{P}$ rotates $v$ counterclockwise about the axis $\left[ {\begin{array}{*{20}{c}} 1 \ 1 \ 1 \end{array}} \right]$ by $90^\circ$.

The other options are incorrect because they are not angles of rotation.