Linear Algebra MCQs and MSQs with Solutions for CSIR NET, IIT JAM : Bilinear and Quadratic Forms - III

21. Let \(A=\begin{bmatrix}3 & 0 & 0 \\ 0 & 6 & 2 \\ 0 & 2 & 6\end{bmatrix}\) and let \(\lambda_1 \geq \lambda_2 \geq \lambda_3\) be the eigenvalue of \(\mathrm{A}\). The triple \((\lambda_1, \lambda_2, \lambda_3)\) equals

(a.) \((9,4,2)\)
(b.) \((8,4,3)\)
(c.) \((9,3,3)\)
(d.) \((7,5,3)\)

22. \(A=\begin{bmatrix}3 & 0 & 0 \\ 0 & 6 & 2 \\ 0 & 2 & 6\end{bmatrix}\) The matrix \(P\) such that \(P^{\prime} A P=\begin{bmatrix}\lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_2\end{bmatrix}\) is

(a.) \(\begin{bmatrix}\frac{1}{\sqrt{3}} & 0 & \frac{-2}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}}\end{bmatrix}\)
(b.) \(\begin{bmatrix}\frac{1}{\sqrt{3}} & \frac{-2}{\sqrt{6}} & 0 \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{6}} & -\frac{1}{\sqrt{2}}\end{bmatrix}\)
(c.) \(\begin{bmatrix}0 & 0 & 0 \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0\end{bmatrix}\)
(d) \(\begin{bmatrix}0 & 1 & 0 \\ \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}}\end{bmatrix}\)

23. Let \(V\) be a vector space (over \(R\)) of dimension 7 and let \(f: V \rightarrow \mathbf{R}\) be a non-zero linear functional. Let \(W\) be a linear subspace of \(V\) such that \(V=\operatorname{Ker}(f) \oplus W\) where \(\operatorname{Ker}(f)\) is the null space of \(f\). What is the dimension of \(W\)?

(a.) 0
(b.) 1
(c.) 6
(d.) None of these

24. Consider the following statements:

1. Let \(A\) be a hermitian \(N \times N\) positive definite matrix. Then, there exists a hermitian positive definite \(N \times N\) matrix \(B\) such that \(B^2=A\).

2. Let \(B\) be a nonsingular \(N \times N\) matrix with real entries. Let \(B^{\prime}\) be its transpose. Then \(B^{\prime} B\) is a symmetric and positive definite matrix.

(a.) 1 true 2 false
(b.) 2 true 1 false
(c.) Boht are true
(d.) Both are false

25. Let \(A\) be a symmetric \(n \times n\) matrix with real entries, which is positive semi-definite, i.e., \(x^T A x \geq 0\) for every (column) vector \(x\), where \(x^T\) denotes the (row) vector which is the transpose of \(x\). Pick out the true statements:

(a.) The eigenvalues of \(A\) are all nonn egative
(b.) \(A\) is invertible;
(c.) The principal minor \(\Delta_k\) of \(A\) (i.e. the determinant of the \(k \times k\) matrix obtained from the first \(k\) rows and first \(k\) columns of \(A\)) is non-negative for each \(1 \leq k \leq n\).
(d.) None of these

26. Consider the matrix \(M=\begin{bmatrix}0 & 1 & 2 & 0 \\ 1 & 0 & 1 & 0 \\ 2 & 1 & 0 & 2 \\ 0 & 0 & 2 & 0\end{bmatrix}\). Then

(a.) M has no real eigen values
(b.) All real eigen values of \(M\) are positive
(c.) All real eigen values of \(\mathrm{M}\) are negative
(d.) \(\mathrm{M}\) has both positive and negative real eigen values

27. Let \(f\) be a non-zero symmetric bilinear from on \(\mathbb{R}^3\). suppose that exist linear transformations \(T_i: \mathbb{R}^3 \rightarrow \mathbb{R}, i=1,2\) such that for all \(\alpha, \beta \in \mathbb{R}^3, f(\alpha, \beta)=T_1(\alpha) T_2(\beta)\).

(a.) Rank \(f=1\)
(b.) \(\operatorname{dim} \quad\left\{\beta \in \mathbb{R}^3: f(\alpha, \beta)=0 \quad\right.\) for all \(\left.\alpha \in \mathbb{R}^3\right\}=2\)
(c.) \(f\) is a positive semi-definite or negative semi-definite
(d.) \(\{\alpha: f(\alpha, \alpha)=0\}\) is a linear subspace of dimension 2.

28. Let \(A=\begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}\) and let \(\alpha_n\) and \(\beta_n\) denote the two eigenvalues of \(A^n\) such that \(\left|\alpha_n\right| \geq\left|\beta_n\right|\). Then

(a.) \(\alpha_n \rightarrow \infty\) as \(n \rightarrow \infty\)
(b.) \(\beta_n \rightarrow 0\) as \(n \rightarrow \infty\)
(c.) \(\beta_n\) is positive if \(n\) is even.
(d.) \(\beta_n\) is negative if \(n\) is odd.

29. Let \(A\) be a \(n \times n\) real symmetric non-singular matrix. Suppose there exists \(r \in \mathbb{R}^n\) such that \(x^{\prime} A x<0\) Then we can conclude that

(a.) \(\operatorname{det}(A)<0\)
(b.) \(B=-A\) is positive definite.
(c.) \(\exists y \in \mathbb{R}^n: y^{\prime} A^{-1} y<0\)
(d.) \(\forall y \in \mathbb{R}^n: y^{\prime} A^{-1} y<0\)

30. Let \(A=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}\). Let \(f: \mathbb{R}^2 \times \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(v, w)=w^T\) Av Pick the correct statement form below:

(a.) There exists and eigenvector \(v\) of \(A\) such that \(A v\) is perpendicular to \(v\)
(b.) The set \(\left\{v \in \mathbb{R}^2 \mid f(v, v)=0\right\}\) nonvero subspace of \(\mathbb{R}^2\)
(c.) If \(v, w \in \mathbb{R}^2\) are nonzero vectors such that \(f(v, v)=0=f(w, w)\), then \(v\) is a scalar multiple of \(w\)
(d.) For every \(v \in \mathbb{R}^2\), there exists a nonzero we \(\mathrm{R}^2\) such that \(f(x, w)=0\)

31. The matrix

\(\begin{bmatrix}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{bmatrix}\) is

(a.) Positive-definite
(b.) Non-negative definite but not positive definite
(c.) Negative definite
(d.) Neither negative definite nor positive definite