Consider the following in respect of the matrices P=\begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix} and Q=\begin{bmatrix} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{bmatrix} I. PQ is a null matrix. II. QP is an identity matrix of order 3. III. PQ=QP Which of the above is/are correct?

Explanation

Matrix P corresponds to the cross-product operation with vector \vec{v}=(a,b,c), and matrix Q represents the outer product \vec{v}\vec{v}^T. Evaluating PQ on a vector \vec{x} computes \vec{v} \times ((\vec{v}\cdot\vec{x})\vec{v}), which is \vec{0} because \vec{v} \times \vec{v} = \vec{0}. Thus, PQ is a null matrix. Similarly, QP = \vec{0}. This means PQ = QP = 0, validating statements I and III, while II is incorrect.

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