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

41. Let \(A=\left(a_{ij}\right)\) be a non-zero matrix of order 8 with \(A^2=0\). Which of the following is the possible rank of \(A\)?

(a.) 5
(b.) 6
(c.) 4
(d.) 8

42. A basis of the eigenspace of the matrix \(\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\) consists of

(a.) \(\begin{bmatrix}1 \\ 1\end{bmatrix}\)
(b.) \(\begin{bmatrix}0 \\ 0\end{bmatrix}\) and \(\begin{bmatrix}1 \\ 0\end{bmatrix}\)
(c.) \(\begin{bmatrix}0 \\ 1\end{bmatrix}\)
(d.) \(\begin{bmatrix}0 \\ 1\end{bmatrix}\) and \(\begin{bmatrix}1 \\ 0\end{bmatrix}\)

43. If \(\begin{bmatrix}1 \\ -1\end{bmatrix}\) is an eigenvector of \(A\), then \(n\) is

(a.) 1
(b.) 2
(c.) 3
(d.) None of these

44. The characteristic roots of the \(3 \times 3\) matrix \(A\) are 3, 2, and -1. If \(B=A^2-A\), then the determinant of \(B\) is

(a.) 24
(b.) -2
(c.) 12
(d.) -15

45. Let \(A\) be a real \(4 \times 4\) matrix with characteristic polynomial \(P(T)=(T^2+1)^2\). Which of the following is true?

(a.) \(A\) is diagonalizable over complex numbers but not over real numbers
(b.) \(A\) is nilpotent
(c.) \(A\) is invertible
(d.) There is no such matrix \(A\)

46. If the characteristic roots of \(\begin{bmatrix}3 & 7 \\ 2 & 5\end{bmatrix}\) are \(\lambda_1\) and \(\lambda_2\), the characteristic roots of \(\begin{bmatrix}5 & -7 \\ -2 & 3\end{bmatrix}\) are

(a.) \(\lambda_1, \lambda_2, \lambda_1-\lambda_2\)
(b.) \(\lambda_1\) and \(\lambda_2\)
(c.) \(2\lambda_1\) and \(2\lambda_2\)
(d.) \(\lambda_1+\lambda_2\) and \(\left|\lambda_1-\lambda_2\right|\)

47. If \(A=\begin{bmatrix}2 & 1 \\ 3 & -1\end{bmatrix}\) and \(I=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\), which of the following is the zero matrix?

(a.) \(A^2+A-5I\)
(b.) \(A^2+A-5I\)
(c.) \(A^2+A-I\)
(d.) \(A^2+3A-5I\)

48. If \(T: V \rightarrow V\) is a linear operator for \(\operatorname{Dim} V=n\) and \(T\) has distinct eigenvalues, then

(a.) \(T\) must be invertible
(b.) \(T\) must be diagonalizable
(c.) \(T\) must be invertible as well as diagonalizable
(d.) \(T\) is not diagonalizable

49. The matrix \(\begin{bmatrix}2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\) is similar to the matrix

(a.) \(\begin{bmatrix}1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{bmatrix}\)
(b.) \(\begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 3\end{bmatrix}\)
(c.) \(\begin{bmatrix}0 & 1 & 0 \\ -1 & 4 & 0 \\ 0 & 0 & 0\end{bmatrix}\)
(d.) \(\begin{bmatrix}3 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & -4 & 4\end{bmatrix}\)