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

71. Let \(A\) be a \(3 \times 3\) matrix. Suppose that the eigenvalues of \(A\) are \(-1,0,1\) with respective eigenvectors \((1,-1,0)^t\), \((1,1,-2)^t\) and \((1,1,1)^t\). Then \(6A\) equals

(a.) \(\begin{bmatrix}-6 & 30 & 12 \\ 30 & -6 & 12 \\ 12 & 12 & 12\end{bmatrix}\)
(b.) \(\begin{bmatrix}6 & 0 & 0 \\ 0 & -6 & 0 \\ 0 & 0 & 0\end{bmatrix}\)
(c.) \(\begin{bmatrix}6 & 30 & 18 \\ 30 & 6 & 18 \\ 18 & 18 & 18\end{bmatrix}\)
(d.) \(\begin{bmatrix}-18 & 54 & 0 \\ 54 & -18 & 0 \\ 0 & 0 & 36\end{bmatrix}\)

72. Let \(M=\begin{bmatrix}1 & a & b \\ 0 & 2 & c \\ 0 & 0 & 1\end{bmatrix}\), \(a, b, c \in \mathbb{R}\). Then, \(M\) is diagonalizable if and only if

(a.) \(a=bc\)
(b.) \(b=ac\)
(c.) \(c=ab\)
(d.) \(a=b=c\)

73. Let \(M\) be the real \(5 \times 5\) matrix having all of its entries equal to 1. Then,

(a.) \(M\) is not diagonalizable
(b.) \(M\) is idempotent
(c.) \(M\) is nilpotent
(d.) The minimal polynomial and the characteristic polynomial of \(M\) are not equal

74. Let \(M=\begin{bmatrix}1 & 3 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 9\end{bmatrix}\). Then

(a.) \(M\) is diagonalizable but not \(M^2\)
(b.) \(M^2\) is diagonalizable but not \(M\)
(c.) Both \(M\) and \(M^2\) are diagonalizable
(d.) Neither \(M\) nor \(M^2\) is diagonalizable

75. Let \(M\) be a skew-symmetric, orthogonal real matrix, The only possible eigenvalues are

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

76. If a square matrix of order 10 has exactly 4 distinct eigenvalues, then the degree of its minimal polynomial is

(a.) At least 4
(b.) At most 4
(c.) At least 6
(d.) At most 6

77. Let \(P\) be a \(n \times n\) matrix with integral entries and \(Q=P+\frac{1}{2} I\), where \(I\) denotes the \(n \times n\) identity matrix. Then \(Q\) is

(a.) Idempotent, i.e., \(Q^2=Q\)
(b.) Invertible
(c.) Nilpotent
(d.) Unipotent, i.e., \(Q-I\) is nilpotent

78. Let \(M\) be a square matrix of order 2 such that the rank of \(M\) is 1. Then \(M\) is

(a.) Diagonalizable and nonsingular
(b.) Diagonalizable and nilpotent
(c.) Neither diagonalizable nor nilpotent
(d.) Either diagonalizable or nilpotent but not both

79. Let the characteristic equation of a matrix \(M\) be \(\lambda^2-\lambda-1=0\), then

(a.) \(M^{-1}\) does not exist
(b.) \(M^{-1}\) exists but cannot be determined
(c.) \(M^{-1}=M+1\)
(d.) \(M^{-1}=M-1\)

80. Consider the matrix \(M=\begin{bmatrix}1 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 1 & -1\end{bmatrix}\) and let \(S_M\) be the set of \(3 \times 3\) matrices \(N\) such that \(M \cdot N=0\). Then the dimension of the real vector space \(S_M\) is equal to

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