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

81. Consider the matrix \(M=\begin{bmatrix}a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2\end{bmatrix}\) Then the matrix has

(a.) Three non-zero real eigenvalues
(b.) Complex eigenvalues
(c.) Two non-zero eigenvalues
(d.) Only one non-zero eigenvalue

82. Let \(A\) be a non-zero upper triangular matrix all of whose eigenvalues are 0. Then \(I+A\)

(a.) Invertible
(b.) Singular
(c.) Idempotent
(d.) Nilpotent

83. The eigenvalues of a skew-symmetric matrix

(a.) Negative
(b.) Real
(c.) Of an absolute value 1
(d.) Purely imaginary or zero

84. Let \(A\) be a \(3 \times 3\) matrix with eigenvalues \(1, -1, 0\). Then the determinant of \(I+A^{100}\) is

(a.) 6
(b.) 4
(c.) 9
(d.) 100

85. The eigenvalues of a \(3 \times 3\) real matrix \(P\) are

(a.) \(P^{-1}=\frac{1}{6}(5I+2P-P^2)\)
(b.) \(P^{-1}=\frac{1}{6}(5I-2P+P^2)\)
(c.) \(P^{-1}=\frac{1}{6}(5I-2P-P^2)\)
(d.) \(P^{-1}=\frac{1}{6}(5I+2P+P^2)\)

86. Let \(-T: \mathbb{C}^n \rightarrow \mathbb{C}^n\) be a linear operator having \(n\)-distinct eigen values. Then

(a.) \(T\) is invertible
(b.) \(T\) is invertible as well as diagonalizable
(c.) \(T\) is not diagonalizable
(d.) \(T\) is diagonalizable

87. Let \(U\) be a \(3 \times 3\) complex Hermitian matrix which is Unitary. Then the distinct eigenvalues of \(U\) are

(a.) \(\pm i\)
(b.) \(1 \pm i\)
(c.) \(\pm 1\)
(d.) \(\frac{1}{2}(1 \pm i)\)

88. Let \(A\) be an \(n \times n\) complex matrix whose the characteristic polynomial is given by \(f(t)=t^n+c_{n-1} t^{n-1}+\ldots . .+c_1 t+c_0\). Then

(a.) \(\operatorname{det}(A)=c_{n-1}\)
(b.) \(\operatorname{det}(A)= c_0\)
(c.) \(\operatorname{det}(A) =(-1)^n c_{n-1}\)
(d.) \(\operatorname{det}(A)=(-1)^n c_0\)

89. Let \(T: C^n \rightarrow C^n\) be a linear operator rank \(n-2\). Then

(a.) 0 is not an eigenvalue of \(T\)
(b.) 0 must be an eigenvalue of \(T\)
(c.) 1 can never be an eigenvalue of \(T\)
(d.) 1 must be an eigenvalue of \(T\)

90. For \(0<\theta<\pi\), the matrix \(\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}\)

(a.) Has no real eigenvalue
(b.) Is orthogonal
(c.) Is symmetric
(d.) Is skew-symmetric

91. Let \(P\) and \(Q\) be square matrices such that \(P Q=I\), the identity matrix. Then zero is an eigenvalue of

(a.) \(P\) but not of \(Q\)
(b.) \(Q\) but not of \(P\)
(c.) Both \(P\) and \(Q\)
(d.) Neither \(P\) nor \(Q\)

92. Let \(A\) and \(B\) be \(n \times n\) matrices with the same minimal polynomial. Then

(a.) \(A\) is similar to \(B\)
(b.) \(A\) is diagonalizable diagonalizable
(c.) \(A-B\) is singular
(d.) \(A\) and \(B\) commute

93. Let \(A\) be an \(n \times n\) matrix which is both Hermitian and unitary. Then

(a.) \(A^2=I\)
(b.) \(A\) is real
(c.) The eigenvalue of \(A\) are \(0,1,-1\)
(d.) The characteristic and minimal polynomials of \(A\) are the same.

94. A matrix \(M\) has eigen values 1 and 4 with corresponding eigen vectors \((1,-1)^T\) and \((2,1)^T\), respectively. Then \(M\) is

(a.) \(\begin{bmatrix}-4 & -8 \\ 5 & 9\end{bmatrix}\)
(b.) \(\begin{bmatrix}9 & -8 \\ 5 & -4\end{bmatrix}\)
(c.) \(\begin{bmatrix}2 & 2 \\ 1 & 3\end{bmatrix}\)
(d.) \(\begin{bmatrix}3 & 2 \\ 1 & 2\end{bmatrix}\)

95. Let \(A=\left(a_{i j}\right)\) be an \(n \times n\) matrix such that \(a_{i j}=3\) for \(i\) and \(j\). Then the nullity of \(A\) is

(a.) \(n-1\)
(b.) \(n-3\)
(c.) \(n\)
(d.) 0