List of top Engineering Mathematics Questions on Eigenvalues asked in GATE AE

Consider the matrix \[A = \begin{bmatrix} 5 & -4 \\ k & -1 \end{bmatrix},\] 
where \(k\) is a constant. If \(\det(A) = 3\), then the ratio of the largest eigenvalue of \(A\) to \(k\) is ___________ (rounded off to 1 decimal place).
 

Which of the following statement(s) is/are true with respect to eigenvalues and eigenvectors of a matrix?
The largest eigenvalue of the given matrix is: \[ \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{bmatrix} \]