Consider the matrices X(4 × 3), Y(4 × 3) and P(2 × 3). The order of [P(XTY)-1 PT]T will be A. (2 × 2) B. (3 × 3) C. (4 × 3) D. (3 × 4)

The correct answer is (3 × 4).

The order of a matrix is the number of rows in the matrix by the number of columns in the matrix. For example, a matrix with 3 rows and 4 columns would have an order of $(3\times4)$.

In this question, we are given the matrices $X$, $Y$, and $P$, where $X$ is a $(4\times3)$ matrix, $Y$ is a $(4\times3)$ matrix, and $P$ is a $(2\times3)$ matrix.

We are asked to find the order of the matrix $[P(XTY)^{-1}PT]^T$.

To find the order of this matrix, we can first find the order of the matrix $P(XTY)^{-1}$.

The order of $P(XTY)^{-1}$ is the same as the order of $XTY$, which is $(4\times3)$.

The order of $[P(XTY)^{-1}PT]^T$ is the transpose of the order of $P(XTY)^{-1}$, which is $(3\times4)$.

Therefore, the order of $[P(XTY)^{-1}PT]^T$ is $(3\times4)$.

Here is a brief explanation of each option:

• Option A: $(2\times2)$. This is not the correct answer because the order of $[P(XTY)^{-1}PT]^T$ is $(3\times4)$, which is not equal to $(2\times2)$.
• Option B: $(3\times3)$. This is not the correct answer because the order of $[P(XTY)^{-1}PT]^T$ is $(3\times4)$, which is not equal to $(3\times3)$.
• Option C: $(4\times3)$. This is not the correct answer because the order of $[P(XTY)^{-1}PT]^T$ is $(3\times4)$, which is not equal to $(4\times3)$.
• Option D: $(3\times4)$. This is the correct answer because the order of $[P(XTY)^{-1}PT]^T$ is $(3\times4)$.