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

51. Let \(V\) be the vector space \(\mathbb{R}^2\) and let \(T\) be the linear transformation on \(V\) defined by \(T(x, y)=(x+y, y)\). Then the characteristic polynomial of \(T\) is

(a.) \(1+3x+x^2\)
(b.) \(2+2x\)
(c.) \(1+2x+x^2\)
(d.) \(1+x^2\)

52. A \(2 \times 2\) real matrix \(A\) is diagonalizable then:

(a.) \((\text { tr } A)^2 \leq 4 \operatorname{Det} A\)
(b.) \((\operatorname{tr} A)^2 \geq 4 \operatorname{Det} A\)
(c.) \((\operatorname{tr} A)^2 \leq 4 \operatorname{Det} A\)
(d.) \(\operatorname{Tr} A=\operatorname{Det} A\)

53. Let \(A\) be a \(3 \times 3\) complex matrix such that \(A^3=I\) (= the \(3 \times 3\) identity matrix). Then:

(a.) \(A\) is diagnonalizable.
(b.) \(A\) is not diagonalizable.
(c.) The minimal polynomial of \(A\) has repeated root.
(d.) All eigenvalues of \(A\) are real.

54. The minimal polynomial of the \(3 \times 3\) real matrix \(\begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & a & b \end{bmatrix}\) is:

(a.) \((X-a)(X-b)\)
(b.) \((X-a)^2(X-b)\)
(c.) \((X-a)^2(X-b)^2\)
(d.) \((X-a)(X-b)^2\)

55. The characteristic polynomial of the \(3 \times 3\) real matrix \(A=\begin{bmatrix}0 & 0 & -c \\ 1 & 0 & -b \\ 0 & 1 & -a\end{bmatrix}\) is

(a.) \(X^3+aX^2+bX+c\)
(b.) \((X-a)(X-b)(X-c)\)
(c.) \((X-1)(X-abc)^2\)
(d.) \((X-1)^2(X/abc)\)

56. 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

(a.) \(11\)
(b.) \(-11\)
(c.) \(-1\)
(d.) \(1\)

57. 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

(a.) \(1\)
(b.) \(2\)
(c.) \(3\)
(d.) \(0\)

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

(a.) \(6\)
(b.) \(3\)
(c.) \(9\)
(d.) \(1\)

59. Let \(T: \mathbb{R}^4 \rightarrow \mathbb{R}^4\) be a linear map such that the null space of \(T\) is \(\{(x, y, z, w) \in \mathbb{R}^4: x+y+z+w=0\}\) and the rank of \((T-4I_4)\) is 3. If the minimal polynomial of \(T\) is \(x(x-4)^\alpha\), then \(\alpha\) is equal to

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

60. Let \(M\) be an invertible Hermitian matrix and let \(x, y \in \mathbb{R}\) be such that \(x^2<4y\). Then

(a.) both \(M^2+xM+yI\) and \(M^2-xM+yI\) are singular
(b.) \(M^2+xM+yI\) is singular but \(M^2-xM+yI\) is non-singular
(c.) \(M^2+xM+yI\) is non-singular but \(M^2-xM+yI\) is singular
(d.) both \(M^2+xM+yI\) and \(M^2-xM+yI\) are non-singular