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

31. Let \(V\) be the vector space of polynomials of degree \(\leq 5\) over \(\mathbb{R}\). Let \(D: V \rightarrow V\) be the derivative map \(p \rightarrow p^{\prime}\). Then

(a.) 0 is the only eigenvalue of \(D\).
(b.) 1 is an eigenvalue of \(D\).
(c.) 5 is an eigenvalue of \(D\).
(d.) \(D\) is invertible.

32. If \(A\) is a real \(2 \times 2\) matrix such that \(A^2-A=0\), then

(a.) Either \(A=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\) or \(A=\begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}\).
(b.) There are infinitely many such matrices \(A\).
(c.) There are only finitely many such matrices \(A\).
(d.) \(A\) has to be a diagonal matrix.

33. Given that a \(3 \times 3\) matrix satisfies the equation \(A^3-A^2+A-I=0\), the value of \(A^4\) is

(a.) Not computable from the given data.
(b.) \(-A^3+A^2-A-I\).
(c.) 0.
(d.) I.

34. Let \(A\) be the \(n \times n\) matrix with all entries equal to 1. The eigenvalues of \(A\) are

(a.) 0 with multiplicity \((n-1)\) and \(n\) with multiplicity 1.
(b.) 0 with multiplicity 1 and \(n\) with multiplicity \((n-1)\).
(c.) 0 with multiplicity 1 and 1 with multiplicity 1.
(d.) \(n\) with multiplicity 1 and 1 with multiplicity 1.

35. A square matrix \(A\) is said to be idempotent if \(A^2=A\). An idempotent matrix is singular if and only if

(a.) All eigenvalues are real.
(b.) All eigenvalues are non-negative.
(c.) All eigenvalues are either 1 or 0.
(d.) All eigenvalues are 1.

36. Let \(A\) be a symmetric idempotent matrix. Which of the following is not true

(a.) \(\operatorname{det}(A) \neq 0 \Rightarrow A= I\).
(b.) Trace \(A=\) Rank \(A\).
(c.) If the \(i^{th}\) diagonal element of \(A\) is zero, at least one element in the \(i^{th}\) is non-zero.
(d.) Every latent/characteristic root of \(A\) is either 0 or 1.

37. Let \(A=\begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{bmatrix}\). Then the eigenvalues of \(A\) are

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

38. Let \(A=\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}\) be such that \(A\) has real eigenvalues. Then

(a.) \(\theta=n\pi\) for some integer \(n\).
(b.) \(\theta=2n\pi+\frac{\pi}{2}\) for some integer \(n\).
(c.) There is no restriction on \(\theta\).
(d.) \(\theta=2n\pi+\frac{\pi}{2}\) for some integer \(n\).

39. \(A=\begin{bmatrix}a_{i j}\end{bmatrix}\) be an \(n \times n\) matrix with real entries such that the sum of all entries in each row is zero. Consider the following statements:

A. \(A\) is non-singular

B. \(A\) is singular

C. 0 is an eigenvalue of \(A\)

Which of the following is correct?

(a.) Only A is true.
(b.) A and C are true.
(c.) B and C are true.
(d.) Only C is true.

40. Let \(A=\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}\). Then the minimal polynomial of \(A\) is

(a.) \(\lambda^2-4 \lambda-5\).
(b.) \(\lambda^2-5 \lambda-4\).
(c.) \(\lambda^2-3 \lambda^2-9 \lambda-5\).
(d.) \(\lambda^3+3 \lambda^2-9 \lambda+5\).