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

21. Suppose the matrix \(A=\begin{bmatrix}40 & -29 & -11 \\ -18 & 30 & -12 \\ 26 & 24 & -50\end{bmatrix}\) has a certain complex number \(\lambda \neq 0\) as an eigenvalue. Which of the following numbers must also be an eigenvalue of \(A\)?

(a.) \(\lambda+20\).
(b.) \(\lambda-20\).
(c.) \(20-\lambda\).
(d.) \(-20-\lambda\).

22. Let \(A\) be a \(3 \times 3\) matrix with real entries such that \(\operatorname{det}(A)=6\) and the trace of \(A\) is 0 . If \(\operatorname{det}(A+I)=0\), where \(I\) denotes the \(3 \times 3\) identity matrix, then the eigenvalues of \(A\) are

(a.) \(-1,2,3\).
(b.) \(-1,2,-3\).
(c.) \(1,2,-3\).
(d.) \(-1,-2,3\).

23. Consider a matrix \(A=\left(a_{i j}\right)_{n \times n}\) with integer entries such that \(a_{i j}=0\) for \(i>j\) and \(a_{i i}=1\) for \(i=1, \ldots, n\). Which of the following properties must be true?

(a.) \(A^{-1}\) exists and it has integer entries.
(b.) \(A^{-1}\) exists and it has some entries that are not integers.
(c.) \(A^{-1}\) is a polynomial function of \(A\) with integer coefficients.
(d.) \(A^{-1}\) is not a power of \(A\) unless \(A\) is the identity matrix.

24. Let \(\omega\) be a complex number such that \(\omega^3=1\), but \(\omega \neq 1\). If

$$A=\begin{bmatrix}1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & \omega & 1\end{bmatrix},$$

Then which of the following statements are true?

(a.) \(A\) is invertible.
(b.) \(\operatorname{rank}(A)=2\).
(c.) 0 is an eigenvalues of \(A\).
(d.) There exist linearly independent vectors \(v, w \in \mathbb{C}^3\) such that \(A v=A w=0\).

25. Let \(A\) be a \(4 \times 4\) matrix with real entries such that \(-1,1,2,-2\) are its eigenvalues. If \(B=A^4-5 A^2+5 I\), where \(I\) denotes the \(4 \times 4\) identity matrix, then which of the following statements are correct?

(a.) \(\operatorname{det}(A+B)=0\).
(b.) \(\operatorname{det}(B)=1\).
(c.) trace of \(A-B\) is 0.
(d.) trace of \(A+B\) is 4.

26. Let \(M_2(\mathbb{R})\) denote the set of \(2 \times 2\) real matrices. Let \(A \in M_2(\mathbb{R})\) be of trace 2 and determinant -3. Identifying \(M_2(\mathbb{R})\) with \(\mathbb{R}^4\), consider the linear transformation \(T: M_2(\mathbb{R}) \rightarrow M_2(\mathbb{R})\) defined by \(T(B)=A B\). Then which of the following statements are true?

(a.) \(T\) is diagonalizable.
(b.) 2 is an eigenvalue of \(T\).
(c.) \(T\) is invertible.
(d.) \(T(B)=B\) for some \(0 \neq B\) in \(M_2(\mathbb{R})\).

27. Let \(A\) be a \(2 \times 2\) non-zero matrix with entries in \(\mathbb{C}\) such that \(A^2=0\). Which of the following statements must be true?

(a.) \(P A P^{-1}\) is diagonal for some invertible \(2 \times 2\) matrix \(P\) with entries in \(\mathbb{R}\).
(b.) \(A\) has two distinct eigenvalues in \(\mathbb{C}\).
(c.) \(A\) has only one eigenvalue in \(\mathbb{C}\) with multiplicity 2.
(d.) \(A v=v\) for some \(v \in \mathbb{C}^2\), \(v \neq 0\).

28. Let \(N\) be a \(3 \times 3\) non-zero matrix with the property \(N^3=0\). 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.

29. Suppose \(A\) is an \(n \times n\) real symmetric matrix with eigenvalues \(\lambda_1, \lambda_2, \ldots, \lambda_n\) then

(a.) \(\prod_{i=1}^n \lambda_1<\operatorname{det}(A)\).
(b.) \(\prod_{i=1}^n \lambda_1>\operatorname{det}(A)\).
(c.) \(\prod_{i=1}^n \lambda_1=\operatorname{det}(A)\).
(d.) If \(\operatorname{det}(A)=1\) then \(\lambda_j=1\) for \(j=1, \ldots, n\).

30. Let \(A\) be an \(n \times n\) matrix, \(n \geq 2\), with characteristic polynomial \(x^{n-2}\left(x^2-1\right)\). Then

(a.) \(A^n=A^{n-2}\).
(b.) Rank of \(A\) is 2.
(c.) Rank of \(A\) is at least 2.
(d.) There exist non-zero vectors \(x\) and \(y\) such that \(A(x+y)=x-y\).