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

11. For a fixed positive integer \(n \geq 3\), let \(A\) be the \(n \times n\) matrix defined as \(A=I-\frac{1}{n} J\), where \(I\) is the identity matrix and \(J\) is the \(n \times n\) matrix with all entries equal to 1 . Which of the following statements is NOT true?

(a.) \(A^k=A\) for every positive integer \(k\).
(b.) Trace \((A)=n-1\).
(c.) \(\operatorname{Rank}(A)+\operatorname{Rank}(I-A)=n\).
(d.) \(A\) is invertible.

12. Let \(A\) be an \(n \times n\) matrix with real entries. Which of the following is correct?

(a.) If \(A^2=0\), then \(A\) is diagonalizable over complex numbers.
(b.) If \(A^2=I\), then \(A\) is diagonalizable over real numbers.
(c.) If \(A^2=A\), then \(A\) is diagonalizable only over complex numbers.
(d.) The only matrix of size \(n\) satisfying the characteristic polynomial of \(A\) is \(A\).

13. Let \(A\) be a \(4 \times 4\) invertible real matrix. Which of the following is NOT necessarily true?

(a.) The rows of \(A\) form a basis of \(\mathbb{R}^4\).
(b.) Null space of \(A\) contains only the 0 vector.
(c.) \(A\) has 4 distinct eigenvalues.
(d.) Image of the linear transformation \(x \mapsto A x\) on \(\mathbb{R}^4\) is \(\mathbb{R}^4\).

14. Let \(A \in M_{10}(\mathbb{C})\), the vector space of \(10 \times 10\) matrices with entries in \(\mathbb{C}\). Let \(W_A\) be the subspace of \(M_{10}(\mathbb{C})\) spanned by \(\left\{A^n \mid n \geq 0\right\}\). Choose the correct statements

(a.) For any \(A\), \(\operatorname{dim}\left(W_A\right) \leq 10\).
(b.) For any \(A\), \(\operatorname{dim}\left(W_A\right)<10\).
(c.) For some \(A\), \(10<\operatorname{dim}\left(W_A\right)<100\).
(d.) For some \(A\), \(\operatorname{dim}\left(W_A\right)=100\).

15. Let \(A\) be a complex \(3 \times 3\) matrix with \(A^3=-I\). Which of the following statements are correct?

(a.) \(A\) has three distinct eigenvalues.
(b.) \(A\) is diagonalizable over \(\mathbb{C}\).
(c.) \(A\) is triangularizable over \(\mathbb{C}\).
(d.) \(A\) is non-singular.

16. Let \(f(x)\) be the minimal polynomial of the \(4 \times 4\) matrix \(A=\begin{bmatrix}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{bmatrix}\). Then the rank of the \(4 \times 4\) matrix \(f(A)\) is

(a.) 0.
(b.) 1.
(c.) 2.
(d.) 4.

17. Let \(n\) be an integer \(\geq 2\) and let \(M_n(\mathbb{R})\) denote the vector space of \(n \times n\) real matrices. Let \(B \in M_n(\mathbb{R})\) be an orthogonal matrix and let \(B^t\) denote the transpose of \(B\). Consider \(W_B=\left\{B^t A B: A \in M_n(\mathbb{R})\right\}\). Which of the following are necessarily true?

(a.) \(W_B\) is a subspace of \(M_n(\mathbb{R})\) and \(W_B \leq \operatorname{rank}(B)\).
(b.) \(W_B\) is a subspace of \(M_n(\mathbb{R})\) and \(\operatorname{dim} W_B=\operatorname{rank}(B)_{\operatorname{rank}}(B^t)\).
(c.) \(W_B=M_n(\mathbb{R})\).
(d.) \(W_B\) is not a subspace of \(M_n(\mathbb{R})\).

18. Let \(A\) be a non-zero linear transformation on a real vector space \(V\) of dimension \(n\). Let the subspace \(V_{\circ} \subset V\) be the image of \(V\) under \(A\). Let \(k=\operatorname{dim} V_0

(a.) \(\lambda=1\).
(b.) \(\operatorname{Det} A=|\lambda|^n\).
(c.) \(\lambda\) is the only eigenvalue of \(A\).
(d.) There is a nontrivial subspace \(V_t \subset V\) such that \(Ax=0\) for all \(x \in V_1\).

19. Let \(C\) be an \(n \times n\) real matrix. Let \(W\) be the vector space spanned by \(\left\{I, C, C^2, \ldots, C^{2n}\right\}\). The dimension of the vector space \(W\) is

(a.) \(2n\).
(b.) At most \(n\).
(c.) \(n^2\).
(d.) At most \(2n\).

20. Let \(N\) be a nonzero \(3 \times 3\) matrix with the property \(N^2=0\). which of the following is/are true?

(a.) \(N\) is not similar to a diagonal matrix.
(b.) \(N\) is similar to a diagonal matrix.
(c.) \(N\) has one non-zero eigenvector.
(d.) \(N\) has three linearly independent eigenvectors.