Discussion
Let A and B be real \(n \times n\) matrices. Which of the following statements is false ?
Answer: D
The false statement is if both A and B have the same rank, then they are congruent.
For real symmetric matrices, Sylvester's law of inertia says congruence is determined by the rank and the signature (the numbers of positive and negative eigenvalues). Equal rank is not enough: \(I_{2}\) and \(-I_{2}\) both have rank 2, but \(P^{t}I_{2}P\) is positive semi-definite for every P, so it can never equal \(-I_{2}\).
The others are true. If \(B = P^{t}AP\), then transposing gives \(B^{t} = P^{t}A^{t}P\), and inverting gives \(B^{-1} = Q^{t}A^{-1}Q\) with \(Q = (P^{t})^{-1}\); and congruence is an equivalence relation, so two matrices congruent to I are congruent to each other.
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