Matrix Algebra - General Aptitude - General Aptitude (GA) Previous Year Questions

The value of \(p\) such that the vector \(\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\) is an eigenvector of the matrix \(\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{bmatrix}\) is ______.

Two sequences \([a, b, c]\) and \([A, B, C]\) are related as, \[\begin{bmatrix} A \\ B \\ C \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & W_3^{-1} & W_3^{-2} \\ 1 & W_3^{-2} & W_3^{-4} \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} \quad \text{where } W_3 = e^{j\frac{2\pi}{3}}.\] If another sequence \([p, q, r]\) is derived as, \[\begin{bmatrix} p \\ q \\ r \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & W_3^1 & W_3^2 \\ 1 & W_3^2 & W_3^4 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & W_3^1 & 0 \\ 0 & 0 & W_3^2 \end{bmatrix} \begin{bmatrix} A/3 \\ B/3 \\ C/3 \end{bmatrix},\] then the relationship between the sequences \([p, q, r]\) and \([a, b, c]\) is

The value of x for which all the eigen-values of the matrix given below are real is
\[ \begin{bmatrix} 10 & 5+j & 4 \\ x & 20 & 2 \\ 4 & 2 & -10 \end{bmatrix} \]

Let \( M^4 = I \), (where I denotes the identity matrix) and M  ,   \( M^2 \)  , and \( M^3 \)  . Then, for any natural number k, \( M^{-1} \) equals:

Consider a two-port network with the transmission matrix: \( T = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \). If the network is reciprocal, then