Which of the following statements are correct ?
(A) If \( \mathrm{A}=\begin{bmatrix} 1 & 4\\ 3 & 7\end{bmatrix} \), then A - AT is skew-symmetric matrix.
(B) If A and B are square matrices of equal order and B is symmetric, then AT BA is symmetric.
(C) If A is a square matrix, then AT A is symmetric.
(D) Every element of the principal diagonal of a symmetric matrix is zero.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
For the given matrix:
\( \displaystyle \mathrm{A}^{T}=\begin{bmatrix} 1 & 3\\ 4 & 7\end{bmatrix} \)
\( \displaystyle \mathrm{A}-\mathrm{A}^{T}=\begin{bmatrix} 0 & 1\\ -1 & 0\end{bmatrix} \)
For any square A and symmetric B (\( \mathrm{B}^{T}=\mathrm{B} \)), using \( (\mathrm{PQ})^{T}=\mathrm{Q}^{T}\mathrm{P}^{T} \):
\( \displaystyle \left(\mathrm{A}^{T}\mathrm{B}\mathrm{A}\right)^{T}=\mathrm{A}^{T}\mathrm{B}^{T}\mathrm{A} \)
\( \displaystyle =\mathrm{A}^{T}\mathrm{B}\mathrm{A} \)
\( \displaystyle \left(\mathrm{A}^{T}\mathrm{A}\right)^{T}=\mathrm{A}^{T}\mathrm{A} \)
So \( \mathrm{A}-\mathrm{A}^{T} \) is skew-symmetric, \( \mathrm{A}^{T}\mathrm{B}\mathrm{A} \) and \( \mathrm{A}^{T}\mathrm{A} \) equal their own transposes, and only a skew-symmetric matrix must have a zero diagonal.
Checking the statements
(A) If \( \mathrm{A}=\begin{bmatrix} 1 & 4\\ 3 & 7\end{bmatrix} \), then A - AT is skew-symmetric matrix. — Correct. \( \mathrm{A}-\mathrm{A}^{T} \) has zero diagonal and entries 1 and −1 off it.
(B) If A and B are square matrices of equal order and B is symmetric, then AT BA is symmetric. — Correct. Its transpose is \( \mathrm{A}^{T}\mathrm{B}^{T}\mathrm{A}=\mathrm{A}^{T}\mathrm{B}\mathrm{A} \).
(C) If A is a square matrix, then AT A is symmetric. — Correct. Its transpose is \( \mathrm{A}^{T}\left(\mathrm{A}^{T}\right)^{T}=\mathrm{A}^{T}\mathrm{A} \).
(D) Every element of the principal diagonal of a symmetric matrix is zero. — Incorrect. The identity matrix is symmetric with diagonal entries 1; the zero diagonal belongs to a skew-symmetric matrix.