Question

If n ≥ 2, then which of the following statements is/are true ?

• A. If A and B are n × n real orthogonal matrices such that det(A) + det(B) = 0, then A + B is a singular matrix
• B. If A is an n × n real matrix such that In + A is non-singular, then In + (In + A)-1(In − A) is a singular matrix
• C. If A is an n × n real skew-symmetric matrix, then In - A2 is a non-singular matrix
• D. If A is an n × n real orthogonal matrix, then det(A − λIn) ≠ 0 for all λ ∈ {x ∈ $\R$ ∶ x ≠ ±1}

Answer 1

Answer: (A), (C), (D)

If A and B are n × n real orthogonal matrices such that det(A) + det(B) = 0, then A + B is a singular matrix

Answer 2

The correct option is (A) : If A and B are n × n real orthogonal matrices such that det(A) + det(B) = 0, then A + B is a singular matrix, (C) : If A is an n × n real skew-symmetric matrix, then In - A2 is a non-singular matrix and (D) : If A is an n × n real orthogonal matrix, then det(A − λIn) ≠ 0 for all λ ∈ {x ∈ $\R$ ∶ x ≠ ±1}.