List of top Mathematics Questions on Eigenvalues and Eigenvectors

Match the LIST-I with LIST-II - Let $A = \begin{bmatrix} 1 & 0 \\ 5 & 2 \end{bmatrix}$. Then

Choose the correct answer from the options given below:

Let \(A\) be a \(3\times 3\) real matrix such that given any column vector \(x\in \mathbb{R}^3\), the column vector \(Ax\) is the reflection of \(x\) about the plane

Let \[ P = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix} \in M_5(\mathbb{C}). \]
Which of the following statements is/are TRUE?

Let \( M = (m_{ij}) \) be a \( 3 \times 3 \) real, invertible matrix and \( \sigma \in S_3 \) be the permutation defined by \( \sigma(1) = 2, \sigma(2) = 3 \) and \( \sigma(3) = 1 \). The matrix \( M_\sigma \) is defined by \( n_{ij} = m_{i\sigma(j)} \) for all \( i,j \in \{1, 2, 3\} \). Then, which one of the following is TRUE?

Let \[ M = \begin{pmatrix} 2 & 0 & -1 \\ 4 & 1 & -4 \\ 2 & 0 & x \end{pmatrix} \] for some real number \( x \).
If \( 0 \) is an eigenvalue of \( M \), then \[ (M^4 + M) \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} \] is equal to

Let \[ M = \begin{pmatrix} 6 & 2 & -6 & 8 \\ 5 & 3 & -9 & 8 \\ 3 & 1 & -2 & 4 \end{pmatrix} \] Consider the system \( S \) of linear equations given by:
\[ 6x_1 + 2x_2 - 6x_3 + 8x_4 = 8 \] \[ 5x_1 + 3x_2 - 9x_3 + 8x_4 = 16 \] \[ 3x_1 + x_2 - 2x_3 + 4x_4 = 32 \] where \( x_1, x_2, x_3, x_4 \) are unknowns.
Then, which one of the following is TRUE?

Let \[ M = \begin{pmatrix} 3 & -2 & 0 \\ 2 & 3 & 3 \\ 4 & -1 & x \end{pmatrix} \] for some real number \( x \). Suppose that \( -2 \) and \( 3 \) are eigenvalues of \( M \).
If \[ M^3 \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 125 \\ 125 \end{pmatrix}, \] then which one of the following is TRUE?

Let \( T : P_2(\mathbb{R}) \to P_2(\mathbb{R}) \) be the linear transformation defined by \[ T(p(x)) = p(x + 1), \quad \text{for all } p(x) \in P_2(\mathbb{R}) \] If \( M \) is the matrix representation of \( T \) with respect to the ordered basis \( \{1, x, x^2\} \) of \( P_2(\mathbb{R}) \), then which one of the following is TRUE?