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

To determine which statements are true regarding n × n matrices, we will evaluate each option step-by-step:

1. Statement 1: "If \(A\) and \(B\) are \(n \times n\) real orthogonal matrices such that \(\text{det}(A) + \text{det}(B) = 0\), then \(A + B\) is a singular matrix."
   • Orthogonal matrices have determinants of either +1 or -1. For \(\text{det}(A) + \text{det}(B) = 0\), this implies one could be 1 and the other -1 (e.g., \(\text{det}(A) = 1\) and \(\text{det}(B) = -1\) or vice versa).
   • Thus, if \(A\) and \(B\) are such that \(\text{det}(A) = -\text{det}(B)\), the matrix \(A + B\) will be singular because they would cancel each other out to a degree, causing linear dependence.
2. Statement 2: "If \(A\) is an \(n \times n\) real matrix such that \(I_n + A\) is non-singular, then \(I_n + (I_n + A)^{-1}(I_n - A)\) is a singular matrix."
   • For \(I_n + A\) to be non-singular, it implies \(\text{det}(I_n + A) \neq 0\).
   • If \(I_n + A\) is non-singular, \((I_n + A)^{-1}\) exists.
   • The expression simplifies to \(I_n + (I_n + A)^{-1}(I_n - A) = (I_n + A)^{-1} (2I_n)\), which is non-singular since scalar multiples of invertible matrices are non-singular.
3. Statement 3: "If \(A\) is an \(n \times n\) real skew-symmetric matrix, then \(I_n - A^2\) is a non-singular matrix."
   • For a skew-symmetric matrix \(A\), all eigenvalues are purely imaginary or zero. Therefore, eigenvalues of \(A^2\) are non-positive real numbers (since they are squares of imaginary numbers).
   • Hence, \(A^2\) has all eigenvalues non-positive, making \(I_n - A^2\) have all positive eigenvalues. Thus, \(I_n - A^2\) is non-singular as it has no zero eigenvalues.
4. Statement 4: "If \(A\) is an \(n \times n\) real orthogonal matrix, then \(\text{det}(A - \lambda I_n) \neq 0\) for all \(\lambda \in \{ x \in \mathbb{R} : x \neq \pm 1 \}\)."
   • An orthogonal matrix \(A\) has eigenvalues on the unit circle of the complex plane, meaning possible eigenvalues could be ±1.
   • Thus, for \(\lambda \neq \pm 1\), \(A - \lambda I_n\) has a non-zero determinant, making it non-singular.

The correct answers are statements 1, 3, and 4.