Real matrices [A]3×1, [B]3×3, [C]3×5, [D]5×3, [E]5×5 and [F]5×1 are given. Matrices [B] and [E] are symmetric. Following statements are made with respect to these matrices. 1. Matrix product [F]T[C]T[B] [C] [F] is a scalar. 2. Matrix product [D]T[F] [D] is always symmetric. With reference to above statements, which of the following applies? A. Statement 1 is true but 2 is false B. Statement 1 is false but 2 is true C. Both the statements are true D. Both the statements are false

The correct answer is: A. Statement 1 is true but 2 is false.

Statement 1 is true because the product of a matrix and its transpose is a scalar. This is a property of matrices, and it holds regardless of the specific matrices involved.

Statement 2 is false because the product of a matrix and its transpose is not always symmetric. This is only true if the matrix is symmetric itself. In this case, matrix $D$ is not symmetric, so the product $D^T F D$ is not symmetric.

Here is a more detailed explanation of each statement:

• Statement 1: The product of a matrix and its transpose is a scalar. This is a property of matrices, and it holds regardless of the specific matrices involved. For example, if $A$ is a $3\times3$ matrix and $B$ is a $3\times3$ matrix, then $A^T B$ is a $3\times3$ matrix.

• Statement 2: The product of a matrix and its transpose is not always symmetric. This is only true if the matrix is symmetric itself. A symmetric matrix is a matrix that is equal to its transpose. For example, if $A$ is a $3\times3$ symmetric matrix, then $A^T=A$.

In this case, matrix $D$ is not symmetric. This is because $D$ is a $5\times3$ matrix, and the only way for a $5\times3$ matrix to be symmetric is if it is a diagonal matrix. However, matrix $D$ is not a diagonal matrix, so it is not symmetric.

Therefore, the product $D^T F D$ is not symmetric.