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

1. Let \(S\) be the set of \(3 \times 3\) real matrices \(A\) with \(A^T A=\begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}\). Then the set \(S\) contains -

(a.) a nilpotent matrix.
(b.) a matrix of rank one.
(c.) a matrix of rank two.
(d.) a non-zero skew-symmetric matrix.

2. An \(n \times n\) complex matrix \(A\) satisfies \(A^k=I_n\), the \(n \times n\) identity matrix, where \(k\) is a positive integer \(>1\). Suppose 1 is not an eigenvalue of \(A\). Then which of the following statements are necessarily true?

(a.) \(A\) is diagonalizable.
(b.) \(A+A^2+\ldots+A^{k-1}=0\), The \(n \times n\)-zero matrices.
(c.) \(\operatorname{tr}(A)+\operatorname{tr}\left(A^2\right)+\ldots+\operatorname{tr}\left(A^{k-1}\right)=-n\).
(d.) \(A^{-1}+A^{-2}+\ldots+A^{-(k-1)}=-I_n\).

3. Let \(S: \mathbb{R}^n \rightarrow \mathbb{R}^n\) be given by \(S(v)=\alpha v\) for a fixed \(\alpha \in \mathbb{R}, \alpha \neq 0\). Let \(T: \mathbb{R}^n \rightarrow \mathbb{R}^n\) be a linear transformation such that \(B=\left\{v_1, \ldots, v_n\right\}\) is a set of linearly independent eigenvectors of \(T\). Then

(a.) The matrix of \(T\) with respect to \(B\) is diagonal.
(b.) The matrix of \((T-S)\) with respect to \(B\) is diagonal.
(c.) The matrix of \(T\) with respect to \(B\) is not necessarily diagonal, but is upper triangular.
(d.) The matrix of \(T\) with respect to \(B\) is diagonal, but the matrix of \((T-S)\) with respect to \(B\) is not diagonal.

4. Let \(A=\begin{pmatrix}a & b & c \\ 0 & a & d \\ 0 & 0 & a\end{pmatrix}\) be a \(3 \times 3\) matrix where contains \(a\), \(b\), \(c\), \(d\) are integers. Then, we must have:

(a.) If \(a \neq 0\), there is a polynomial \(p \in \mathbb{Q}[x]\) such that \(p(A)\) is the inverse of \(A\).
(b.) For each polynomial \(q \in \mathbb{Z}[x]\), the matrix \(q(A)=\begin{pmatrix}q(a) & q(b) & q(c) \\ 0 & q(a) & q(d) \\ 0 & 0 & q(a)\end{pmatrix}\).
(c.) If \(A^n=0\) for some positive integer \(n\), then \(A^3=0\).
(d.) \(A\) commutes with every matrix of the form \(\begin{pmatrix}a' & 0 & c' \\ 0 & a' & 0 \\ 0 & 0 & a'\end{pmatrix}\).

5. Let \(A\) be an invertible \(4 \times 4\) real matrix. Which of the following are NOT true?

(a.) Rank \(A=4\).
(b.) For every vector \(b \in \mathbb{R}^4\), \(A x=b\) has exactly one solution.
(c.) \(\operatorname{dim}(\text{nullspace }A) \geq 1\).
(d.) 0 is an eigenvalue of \(A\).

6. Let \(\underline{u}\) be a real \(n \times 1\) vector satisfying \(\underline{u}^{\prime} \underline{u}=1\), where \(\underline{u}^{\prime}\) the transpose of is \(\underline{u}\). Define \(A=I-2 \underline{u} \underline{u}^{\prime}\) where \(I\) is the \(n^{\text {th }}\) order identity matrix. Which of the following statements are true?

(a.) \(A\) is singular.
(b.) \(A^2=A\).
(c.) Trace \((A)=n-2\).
(d.) \(A^2=I\).

7. Let \(P\) be a \(2 \times 2\) complex matrix such that \(p^* p\) is the identity matrix, where \(p^*\) is the conjugate transpose of \(P\). Then the Eigen values of \(P\) are

(a.) real.
(b.) complex conjugates of each other.
(c.) reciprocals of each other.
(d.) of modulus 1.

8. Which of the following are eigenvalues of the matrix \(\begin{pmatrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0\end{pmatrix}\)?

(a.) +1.
(b.) -1.
(c.) \(+i\).
(d.) \(-i\).

9. Let \(J\) denote a \(101 \times 101\) matrix with all the entries equal to 1 and let \(I\) denote the identity matrix of order 101. Then the determinant of \(J-I\) is

(a.) 101.
(b.) 1.
(c.) 0.
(d.) 100.

10. Let \(A\) be a \(5 \times 5\) matrix with real entries such that the sum of the entries in each row of \(A\) is 1. Then the sum of all the entries in \(A^3\) is

(a.) 3.
(b.) 15.
(c.) 5.
(d.) 125.