Final Answer
The transpose of a matrix A formed by concatenating matrices P and Q horizontally over matrices R and S is given by A^T = [P^T Q^T] [R^T S^T].
Step-by-step explanation
The given expression \(A = [P \, Q] [R \, S]\) represents the matrix A derived by horizontally concatenating matrices P and Q over matrices R and S. To compute the transpose of A, denoted as A^T, we transpose each individual matrix within A. The transpose of a matrix involves swapping its rows with columns. Therefore, \(A^T\) will consist of [P^T Q^T] [R^T S^T].
The process of transposing matrices involves flipping their rows and columns. For instance, matrix P^T will have its rows become columns and vice versa. The resulting matrix after transposing P will be P^T. The same process applies to matrices Q, R, and S within matrix A. By transposing each of these matrices individually and maintaining their respective positions within the concatenated structure, we arrive at the expression for the transpose of A as A^T = [P^T Q^T] [R^T S^T].
This process holds due to the nature of matrix operations where the transpose operation is distributive over matrix addition. By applying the transpose operation to each component matrix within A and maintaining the structure of concatenation, the resultant transpose of A is as shown in the equation A^T = [P^T Q^T] [R^T S^T].