List of top Linear Algebra Questions on Eigenvalues and Eigenvectors

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 \(\underline{\hspace{1cm}}\) .

Let \( M \) be an \( n \times n \) ( \( n \ge 2 \) ) non-zero real matrix with \( M^2 = 0 \) and let \( \alpha \in \mathbb{R} \setminus \{0\}. \) Then
Let \( M \) be a real \( 6 \times 6 \) matrix. Let 2 and -1 be two eigenvalues of \( M. \) If \( M^5 = aI + bM \), where \( a, b \in \mathbb{R}, \) then
Let \[ M = \begin{bmatrix} 9 & 2 & 7 & 1 \\ 0 & 7 & 2 & 1 \\ 0 & 0 & 11 & 6 \\ 0 & 0 & -5 & 0 \end{bmatrix}. \] Then, the value of \( \det((8I - M)^3) \) is ................

Let \( I \) denote the \( 4 \times 4 \) identity matrix. If the roots of the characteristic polynomial of a \( 4 \times 4 \) matrix \( M \) are \( \pm \sqrt{\dfrac{1 \pm \sqrt{5}}{2}} \), then \( M^8 \) is