If A and B are square matrices with same order and A is symmetric, then BTAB

Let X = B^T A B. To determine the property of this matrix, we calculate its transpose: X^T = (B^T A B)^T. Using the property (XYZ)^T = Z^T Y^T X^T, we get B^T A^T (B^T)^T. Since A is symmetric, A^T = A. Also, the transpose of a transpose returns the original matrix, so (B^T)^T = B. Substituting these values gives X^T = B^T A B, which is equal to X. Therefore, the matrix B^T A B is symmetric.