Question

If $X_{4\times3}$, $Y_{4\times3}$ and $P_{2\times3}$ are the matrices then the order of the matrix $[P(X^TY)^{-1}P^T]^T$ is

• A. 4x3
• B. 3x4
• C. 3×3
• D. 2×2

Answer 1

Answer: (D)

2×2

Answer 2

Here's a step-by-step breakdown of how to determine the order of the matrix:

1. Dimensions of $X^T$: Since $X$ is a $4 \times 3$ matrix, its transpose $X^T$ will be a $3 \times 4$ matrix.

2. Dimensions of $X^TY$: Multiplying $X^T$ ($3 \times 4$) by $Y$ ($4 \times 3$) results in a $3 \times 3$ matrix.

3. Dimensions of $(X^TY)^{-1}$: The inverse of a $3 \times 3$ matrix (if it exists) is also a $3 \times 3$ matrix.

4. Dimensions of $P(X^TY)^{-1}$: Multiplying $P$ ($2 \times 3$) by $(X^TY)^{-1}$ ($3 \times 3$) results in a $2 \times 3$ matrix.

5. Dimensions of $P^T$: Since $P$ is a $2 \times 3$ matrix, its transpose $P^T$ is a $3 \times 2$ matrix.

6. Dimensions of $P(X^TY)^{-1}P^T$: Multiplying $P(X^TY)^{-1}$ ($2 \times 3$) by $P^T$ ($3 \times 2$) results in a $2 \times 2$ matrix.

7. Dimensions of $[P(X^TY)^{-1}P^T]^T$: The transpose of a $2 \times 2$ matrix is also a $2 \times 2$ matrix.

Therefore, the order of the matrix $[P(X^TY)^{-1}P^T]^T$ is $2 \times 2$.