Linear Algebra · Engineering Mathematics · GATE ECE
Marks 1
Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors
$$ \{ [2 \ -3 \ \alpha], \ [3 \ -1 \ 3], \ [1 \ -5 \ 7] \}$$
does not form a basis of $\mathbb{R}^3$ is _______.
Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvectors of the matrix A, where $$B = {P^{ - 1}}AP$$, respectively, are
Consider a system of linear equations Ax = b, where
$$A = \left[ {\matrix{ 1 \hfill & { - \sqrt 2 } \hfill & 3 \hfill \cr { - 1} \hfill & {\sqrt 2 } \hfill & { - 3} \hfill \cr } } \right]$$, $$b = \left[ {\matrix{ 1 \cr 3 \cr } } \right]$$
This system is equations admits __________.
S1: M has 4 linearly independent eigenvectors.
S2: M has 4 distinct eigenvalues.
S3: M is non-singular (invertible).
Which one among the following is TRUE?
vector $$X = \left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$.
The number of distinct real values of k for which the equation AX = 0 has infinitely many solutions is _______.
It is given that $$A$$ has only one real eigen value. Then the real eigen value of $$A$$ is
Where $$x$$ is unknown. If the eigenvalues of the matrix $$A$$ are $$\left( {\sigma + j\omega } \right)$$ and $$\left( {\sigma - j\omega } \right)$$, then $$x$$ is equal to
The value of $$'k'$$ for which the system has infinitely many solutions is _______.
Which is obtained by reversing the order of the columns of the identity matrix $${{\rm I}_6}$$. Let $$P = {{\rm I}_6} + \alpha \,\,{J_6},$$ where $$\alpha $$ is a non $$-$$ negative real number. The value of $$\alpha $$ for which det $$(P)=0$$ is _______.
$$P = \left[ {\matrix{ {{p_{11}}} & {{p_{12}}} \cr {{p_{21}}} & {{p_{22}}} \cr } } \right]$$ are non-zero and one of the eigen values is zero. Which of the following statement is true?
$${{x_1} + {x_2} + {x_3} = 3}$$
$${{x_1} - {x_3} = 0}$$
$${{x_1} - {x_2} + {x_3} = 1}$$ has
Marks 2
Consider the matrix $\begin{bmatrix}1 & k \\ 2 & 1\end{bmatrix}$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?
Let $$x$$ be an $$n \times 1$$ real column vector with length $$l = \sqrt {{x^T}x} $$. The trace of the matrix $$P = x{x^T}$$ is
The state equation of a second order system is
$$x(t) = Ax(t),\,\,\,\,x(0)$$ is the initial condition.
Suppose $$\lambda_1$$ and $$\lambda_2$$ are two distinct eigenvalues of A and $$v_1$$ and $$v_2$$ are the corresponding eigenvectors. For constants $$\alpha_1$$ and $$\alpha_2$$, the solution, $$x(t)$$, of the state equation is
Let $$\alpha$$, $$\beta$$ two non-zero real numbers and v1, v2 be two non-zero real vectors of size 3 $$\times$$ 1. Suppose that v1 and v2 satisfy $$v_1^T{v_2} = 0$$, $$v_1^T{v_1} = 1$$ and $$v_2^T{v_2} = 1$$. Let A be the 3 $$\times$$ 3 matrix given by :
A = $$\alpha$$v1$$v_1^T$$ + $$\beta$$v2$$v_2^T$$
The eigen values of A are __________.
$$(A)=100$$ and trace $$(A)=14.$$ The value of $$\left| {a - b} \right|$$ is ___________.
The initial conditions are $$x\left[ 0 \right] = 1,\,\,x\left[ 1 \right] = 1$$ and $$x\left[ n \right] = 0$$ for $$n < 0.$$ The value of $$x\left[ {12} \right]$$ is __________.
Eigen value
$${\lambda _1} = 8$$
$${\lambda _2} = 4$$
Eigen vector
$${V_1} = \left[ {\matrix{
1 \cr
1 \cr
} } \right]$$
$${V_2} = \left[ {\matrix{
1 \cr
-1 \cr
} } \right]$$
The matrix is