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:
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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.
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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.