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 I_n + A is non-singular, then I_n + (I_n + A)^-1(I_n − A) is a singular matrix
• C. If A is an n × n real skew-symmetric matrix, then I_n - A² is a non-singular matrix
• D. If A is an n × n real orthogonal matrix, then det(A − λI_n) ≠ 0 for all λ ∈ {x ∈ \(\R\) ∶ x ≠ ±1}

Answer 1

The Correct Option is A, C, D

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 I_n - A² is a non-singular matrix and (D) : If A is an n × n real orthogonal matrix, then det(A − λI_n) ≠ 0 for all λ ∈ {x ∈ \(\R\) ∶ x ≠ ±1}.