Question-11
\(A\) and \(B\) are two square matrices. Select all correct options.
If \(A\) and \(B\) are similar, then we have \(B = P^{-1} A P\) for some invertible matrix \(P\). Multiplying by an invertible matrix doesn’t change the rank, hence the rank of \(A\) and \(B\) are the same. Alternatively, we can view \(A\) and \(B\) as matrix representations of a common linear map \(T\). The rank of \(A\) is equal to the rank of \(T\) which is in turn equal to the rank of \(B\).
Counter example for option-2 \[ A = 3I, B = 5I \] \(A\) and \(B\) have the same rank but are not similar. The only matrix similar to a scalar matrix is the matrix itself.
Counter example for option-3 \[ A = \begin{bmatrix} 2 & 0\\ 0 & 0 \end{bmatrix}, B = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \] \(A\) and \(B\) have the same trace but are not similar. This serves as a counter example for option-4 as well.