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

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