Linear Algebra MCQs and MSQs with Solutions for CSIR NET, IIT JAM : Matrices and Their Properties - VII

61. Let \(A \in M_3(\mathbb{R})\) be such that \(\text{det} (A-I) = 0\) where \(I\) denotes the \(3 \times 3\) identity matrix. If the trace \((A) = 13\) and \(\text{det}(A) = 32\), then the sum of squares of the eigenvalues of \(A\) is___.

(a.) \(194\)
(b.) \(170\)
(c.) \(182\)
(d.) \(206\)

62. The possible set of eigenvalues of a \(4 \times 4\) skew-symmetric orthogonal real matrix is

(a.) \(\{ \pm i\}\)
(b.) \(\{ \pm i, \pm 1\}\)
(c.) \(\{ \pm 1\}\)
(d.) \(\{0, \pm i\}\)

63. Let \(V\) be a vector space of dimension \(m \geq 2\). Let \(T: V \rightarrow V\) be a linear transformation such that \(T^{n+1}=0\) and \(T^n \neq 0\) for some \(n \geq 1\). Then which of the following is necessarily TRUE?

(a.) Rank \((T^n) \leq\) Nullity \((T^n)\)
(b.) Trace \((T) \neq 0\)
(c.) \(T\) is diagonalizable
(d.) \(n=m\)

64. For the matrix \(M=\begin{bmatrix}2 & 3+2i \\ 3-2i & 5 \\ -4 & -6i\end{bmatrix}\) Which of the following statements correct?

P: \(M\) is skew-Hermitian and \(iM\) is Hermitian

Q: \(M\) is Hermitian and \(iM\) is skew-Hermitian

R: eigenvalues of \(M\) are real

S: eigenvalues of \(iM\) are real

(a.) P and R only
(b.) Q and R only
(c.) P and S only
(d.) Q and S only

65. The distinct eigenvalues of the matrix \(\begin{bmatrix}1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}\) are

(a.) \(0\) and \(1\)
(b.) \(1\) and \(-1\)
(c.) \(1\) and \(2\)
(d.) \(0\) and \(2\)

66. The minimal polynomial associated with the matrix \(\begin{bmatrix}0 & 0 & 3 \\ 1 & 0 & 2 \\ 0 & 1 & 1\end{bmatrix}\) is

(a.) \(x^3-x^2-2x-3\)
(b.) \(x^3-x^2+2x-3\)
(c.) \(x^3-x^2-3x-3\)
(d.) \(x^3-x^2+3x-3\)

67. Which of the following matrices is NOT diagonalizable?

(a.) \(\begin{bmatrix}1 & 1 \\ 1 & 2\end{bmatrix}\)
(b.) \(\begin{bmatrix}1 & 0 \\ 3 & 2\end{bmatrix}\)
(c.) \(\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}\)
(d.) \(\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}\)

68. The number of linearly independent eigenvectors of the matrix \(\begin{bmatrix}2 & 2 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 4\end{bmatrix}\) is

(a.) \(1\)
(b.) \(2\)
(c.) \(3\)
(d.) \(4\)

69. Consider the matrix \(N=\begin{bmatrix}3/5 & -4/5 & 0 \\ 4/5 & 3/5 & 0 \\ 0 & 0 & 1\end{bmatrix}\) Then \(N\) is

(a.) Non-invertible
(b.) Skew-symmetric
(c.) Symmetric
(d.) Orthogonal

70. If \(M\) is any \(3 \times 3\) real matrix and \(N=\begin{bmatrix}3/5 & -4/5 & 0 \\ 4/5 & 3/5 & 0 \\ 0 & 0 & 1\end{bmatrix}\), then trace \((N M N^T)\) is equal to

(a.) \([\text{trace}(N)]^2 \text{trace}(M)\)
(b.) \(2\text{trace}(N)+\text{trace}(M)\)
(c.) \(\text{trace}(M)\)
(d.) \([\text{trace}(N)]^2+\text{trace}(M)\)