List of top Linear Algebra Questions on Eigenvalues and Eigenvectors asked in GATE MA

Let \( A \) be a square matrix such that \[ \text{det}(xI - A) = x^4(x - 1)^2(x - 2)^3, \] where \( \text{det}(M) \) denotes the determinant of a square matrix \( M \). If \[ \text{rank}(A^2) < \text{rank}(A^3) = \text{rank}(A^4), \] then the geometric multiplicity of the eigenvalue 0 of \( A \) is \(\underline{\hspace{1cm}}\) .