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

Let T : ℝ⁴ → ℝ⁴ be a linear map defined by
T(x, y, z, w) = (x + z, 2x + y + 3z, 2y + 2z, w).
Then the rank of T is equal to ______

Let \(M\) be a \(3 \times 3\) matrix and suppose that \(1, 2\) and \(3\) are the eigenvalues of \(M\). If \(M^{-1} = \frac{M^2}{\alpha} - M + \frac{11}{\alpha} I_3\) for some scalar \(\alpha \neq 0\), then \(\alpha\) is equal to ________

Let \(M\) be a \(3 \times 3\) singular matrix and suppose that \(2\) and \(3\) are eigenvalues of \(M\). Then the number of linearly independent eigenvectors of \(M^3 + 2M + I_3\) is equal to ________

Let \(M\) be a \(3 \times 3\) matrix such that \(M \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \\ 0 \end{pmatrix}\) and suppose that \(M^3 \begin{pmatrix} 1 \\ -1/2 \\ 0 \end{pmatrix} = \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix}\) for some \(\alpha, \beta, \gamma \in \mathbb{R}\). Then \(|\alpha|\) is equal to ________

For a matrix \(M = [m_{ij}]; \; i,j = 1,2,3,4\), the diagonal elements are all zero and \(m_{ij} = -m_{ji}\). The minimum number of elements required to fully specify the matrix is _____