Discussion
Let A be a 4 × 4 real orthogonal matrix with determinant −1. Then the determinant of I4 − A is :
Answer: A
The determinant of \(I_{4} - A\) is 0.
The eigenvalues of a real orthogonal matrix have modulus 1. The non-real ones come in conjugate pairs \(e^{\pm i\theta}\), each pair contributing 1 to the determinant, and the real ones are \(\pm 1\).
Since \(\det A = -1\), the eigenvalue \(-1\) occurs an odd number of times. In dimension 4 the real eigenvalues are even in number (4 minus twice the number of pairs), so if \(-1\) occurs an odd number of times, \(+1\) must occur an odd number of times as well, at least once. An eigenvalue 1 means \(I - A\) is singular:
\[ \det(I_{4} - A) = 0 \]
For example, A = diag(1, 1, 1, −1) gives \(\det(I - A) = 0 \cdot 0 \cdot 0 \cdot 2 = 0\).
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