Project ForgeWhich of the following statements are TRUE? P. The eigenvalues of a symmetric matrix are real. Q. The value of the determinant of an orthogonal matrix can only be +1. R. The transpose of a square matrix A has the same eigenvalues as those of A. S. The inverse of an 'n×n' matrix exists if and only if the rank is less than 'n'.
- AP and Q only
- P and R only
- CQ and R only
- DP and S only
Explanation
P is true: symmetric matrices always have real eigenvalues. Q is false: an orthogonal matrix's determinant is ±1, not only +1. R is true: det(A−λI)=det(Aᵀ−λI) always, so A and Aᵀ share the same eigenvalues. S is false: an n×n matrix is invertible if and only if its rank EQUALS n (full rank), not less than n. So P and R only.
Source: GATE CH 2013 (2013)
This question is part of a 15-day revision cycle — sign in and Project Forge will bring it back the day before you'd forget it.