If A is any square matrix, then which of the following statements is/are TRUE ?
(A) A + AT is symmetric matrix
(B) A AT is skew symmetric matrix
(C) A - AT is skew symmetric matrix
(D) A2 = (A+I)2 - 2A
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
Each statement is checked with the rules \( (P+Q)^{T}=P^{T}+Q^{T} \) and \( (PQ)^{T}=Q^{T}P^{T} \):
\( (A+A^{T})^{T}=A^{T}+A \)
\( (AA^{T})^{T}=(A^{T})^{T}A^{T}=AA^{T} \)
\( (A-A^{T})^{T}=A^{T}-A=-(A-A^{T}) \)
\( (A+I)^{2}-2A=A^{2}+2A+I-2A \)
\( =A^{2}+I \)
So \( A+A^{T} \) and \( AA^{T} \) are symmetric, \( A-A^{T} \) is skew symmetric, and \( (A+I)^{2}-2A=A^{2}+I \).
Checking the statements
(A) A + AT is symmetric matrix — Correct. Its transpose is \( A^{T}+A \), the same matrix.
(B) A AT is skew symmetric matrix — Incorrect. Its transpose is \( AA^{T} \) itself, so it is symmetric.
(C) A - AT is skew symmetric matrix — Correct. Its transpose is \( -(A-A^{T}) \).
(D) A2 = (A+I)2 - 2A — Incorrect. The right side is \( A^{2}+I \), not \( A^{2} \).