Consider the following in respect of the matrices  \(\rm P=\begin{bmatrix}0&c&-b\\\ -c&0&a\\\ b&-a&0\end{bmatrix}\ and\ \rm Q=\begin{bmatrix}a^2&ab&ac\\\ ab&b^2&bc\\\ ac&bc&c^2\end{bmatrix}\)

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? 

Matrix Multiplication and Properties:

• Matrix multiplication involves the dot product of rows and columns.
• A null matrix is a matrix in which all elements are zero.
• An identity matrix is a square matrix with 1's on the diagonal and 0's elsewhere.
• For matrices P and Q, PQ = QP does not generally hold unless P and Q commute.

Matrix Definitions:

• Null Matrix: A matrix where all elements are zero.
• Identity Matrix: A square matrix with 1's on the main diagonal and 0's elsewhere.

Calculation:

\(\rm P=\begin{bmatrix}0&c&-b\\\ -c&0&a\\\ b&-a&0\end{bmatrix}\ and\ \rm Q=\begin{bmatrix}a^2&ab&ac\\\ ab&b^2&bc\\\ ac&bc&c^2\end{bmatrix}\)

⇒ PQ = \(=\begin{bmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{bmatrix}\ \)

⇒QP = \(=\begin{bmatrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{bmatrix}\ \)

Then PQ = QP

∴ Option (c) is correct