Register

Recent questions tagged matrices

2 2 votes
3 3 answers
1.2k
1.2k views
GATE ECE 2025 | Question: 1
Consider the matrix $A$ below:$$A=\left[\begin{array}{llll}2 & 3 & 4 & 5 \\0 & 6 & 7 & 8 \\0 & 0 & \alpha & \beta \\0 & 0 & 0 & \gamma\end{array}\right]$$For which of the...
User Avatar
0 0 votes
1 1 answer
1.3k
1.3k views
GATE ECE 2024 | Question: 45
​​​​​Consider the matrix $\left[\begin{array}{ll}1 & k \\ 2 & 1\end{array}\right]$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector...
0 0 votes
1 1 answer
1.0k
1.0k views
GATE ECE 2023 | Question: 5
Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_k \mid 1 \leq k \leq n\right\}$ and $\left\{v_k \mid 1 \leq k \leq n\right\}$, respectivel...
0 0 votes
1 1 answer
1.6k
1.6k views
GATE ECE 2023 | Question: 28
Let $\boldsymbol{x}$ be an $n \times 1$ real column vector with length $l=\sqrt{\boldsymbol{x}^{T} \boldsymbol{x}}$ The trace of the matrix $P=\boldsymbol{x} \boldsymbol{...
1 1 vote
0 0 answers
461
461 views
TIFR ECE 2015 | Question: 6
$\textbf{A}$ is an $n \times n$ square matrix of reals such that $\mathbf{A y}=\mathbf{A}^{T} \mathbf{y}$, for all real vectors $\mathbf{y}$. Which of the following can w...
1 1 vote
0 0 answers
412
412 views
TIFR ECE 2014 | Question: 5
The matrix\[A=\left(\begin{array}{ccc}1 & a_{1} & a_{1}^{2} \\1 & a_{2} & a_{2}^{2} \\1 & a_{3} & a_{3}^{2}\end{array}\right)\]is invertible when$a_{1}>a_{2}>a_{3}$$a_{1}...
1 1 vote
0 0 answers
262
262 views
TIFR ECE 2013 | Question: 13
Let $A$ be a square matrix and $x$ be a vector whose dimensions match $A$. Let $B^{\dagger}$ be the conjugate transpose of $B$. Then which of the following is not true:$x...
1 1 vote
0 0 answers
282
282 views
TIFR ECE 2012 | Question: 18
Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the ...
1 1 vote
0 0 answers
429
429 views
TIFR ECE 2012 | Question: 20
Let $A$ be a $2 \times 2$ matrix with all entries equal to $1.$ Define $B=\sum_{n=0}^{\infty} A^{n} / n !$. Then$B=e^{2} A / 2$$B=\left(\begin{array}{cc}1+e & e \\e & 1+e...
1 1 vote
0 0 answers
408
408 views
TIFR ECE 2010 | Question: 10
$\text{H}$ is a circulant matrix (row $n$ is obtained by circularly shifting row $1$ to the right by $n$ positions) and $\text{F}$ is the $\text{DFT}$ matrix. Which of th...
1 1 vote
0 0 answers
410
410 views
TIFR ECE 2010 | Question: 11
Consider\[\text{F}=\frac{1}{2}\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\1 & 1 & -1 & -1 \\1 & -1 & -1 & 1 \\1 & -1 & 1 & -1\end{array}\right], \quad x=\left[\begin{array}{...
1 1 vote
0 0 answers
280
280 views
TIFR ECE 2022 | Question: 12
An $n \times n$ matrix $\mathbf{P}$ is called a Permutation Matrix if each of its $n$ columns and $n$ rows contain exactly one $1$ and $n-1 \; 0$ 's. Consider the followi...
1 1 vote
0 0 answers
293
293 views
TIFR ECE 2020 | Question: 9
Let $A$ be an $n \times n$ matrix with the the property that $A^{m}=0$ for some $m \in \mathbb{N}$. Consider the following statements:At least one entry of $A$ is zeroAll...
1 1 vote
0 0 answers
241
241 views
TIFR ECE 2020 | Question: 14
Two matrices $A$ and $B$ are called similar if there exists an invertible matrix $X$ such that $A=X^{-1} B X$. Let $A$ and $B$ be two similar matrices. Consider the follo...
1 1 vote
0 0 answers
325
325 views
TIFR ECE 2019 | Question: 2
Let $A$ and $B$ be two square matrices that have full rank. Let $\lambda_{A}$ be an eignevalue of $A$ and $\lambda_{B}$ an eigenvalue of $B$. Which of the following is al...
1 1 vote
0 0 answers
385
385 views
TIFR ECE 2018 | Question: 8
Let $A$ be an $n \times n$ real matrix for which two distinct non-zero $n$-dimensional real column vectors $v_{1}, v_{2}$ satisfy the relation $A v_{1}=A v_{2}$. Consider...
1 1 vote
0 0 answers
361
361 views
TIFR ECE 2017 | Question: 13
Let $A$ be an $n \times n$ matrix. Consider the following statements.$A$ can have full-rank even if there exists two vectors $v_{1} \neq v_{2}$ such that $A v_{1}=A v_{2}...
1 1 vote
0 0 answers
308
308 views
TIFR ECE 2016 | Question: 13
Suppose $m$ and $n$ are positive integers, $m \neq n$, and $A$ is an $m \times n$ matrix with real entires. Consider the following statements.$\operatorname{rank}\left(A ...
0 0 votes
0 0 answers
416
416 views
GATE ECE 2004 | Question: 69
If $A=\left[\begin{array}{cc}-2 & 2 \\ 1 & -3\end{array}\right]$, then $\sin A t$ is$\frac{1}{3}\left[\begin{array}{cc}\sin (-4 t)+2 \sin (-t) & -2 \sin (-4 t)+2 \sin (-t...
0 0 votes
0 0 answers
575
575 views
GATE ECE 2005 | Question: 32
Given the matrix $\left[\begin{array}{cc}-4 & 2 \\ 4 & 3\end{array}\right]$, the eigenvector is$\left[\begin{array}{l}3 \\ 2\end{array}\right]$$\left[\begin{array}{l}4 \\...
0 0 votes
0 0 answers
514
514 views
GATE ECE 2005 | Question: 37
Given an orthogonal matrix $A=\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right]$ $\left[\mathrm{AA}^{\mathrm...
3 3 votes
2 2 answers
849
849 views
GATE ECE 2008 | Question: 1
All the four entries of the $2 \times 2$ matrix $\mathbf{P}=\left[\begin{array}{ll}p_{11} & p_{12} \\ p_{21} & p_{22}\end{array}\right]$ are nonzero, and one of its eigen...
2 2 votes
0 0 answers
386
386 views
GATE ECE 2008 | Question: 27
Consider the matrix $\mathbf{P}=\left[\begin{array}{cc}0 & 1 \\ -2 & -3\end{array}\right]$. The value of $\mathbf{e}^{\mathbf{P}}$ is$\left[\begin{array}{cc}2 e^{-2}-3 e^...
0 0 votes
0 0 answers
355
355 views
GATE ECE 1993 | Question 2.7
If $\mathrm{A}=\left(\begin{array}{cccc}1 & 0 & 0 & 1 \\ 0 & -1 & 0 & -1 \\ 0 & 0 & i & i \\ 0 & 0 & 0 & -i\end{array}\right)$the matrix $\mathrm{A}^{4}$, calculated by t...
1 1 vote
1 1 answer
832
832 views
GATE ECE 2022 | Question: 27
Let $\alpha, \beta$ be two non-zero real numbers and $v_{1}, v_{2}$ be two non-zero real vectors of size $3 \times 1.$ Suppose that $v_{1}$ and $v_{2}$ satisfy $v_{1}^{T}...
User Avatar
0 0 votes
1 1 answer
804
804 views
GATE ECE 2019 | Question: 17
The number of distinct eigenvalues of the matrix$$A=\begin{bmatrix} 2&2&3&3\\0&1&1&1\\0&0&3&3\\0&0&0&2 \end{bmatrix}$$is equal to ____________.
User Avatar
0 0 votes
0 0 answers
3.1k
3.1k views
GATE ECE 2016 Set 3 | Question: 1
Consider a $2\times2$ sqaure matrix $$\textbf{A}= \begin{bmatrix} \sigma &x\\ \omega &\sigma \end{bmatrix},$$ where $x$ is unknown. If the eigen values of the matrix $\te...
0 0 votes
0 0 answers
508
508 views
GATE ECE 2016 Set 2 | Question: 1
The value of $x$ for which the matrix $A= \begin{bmatrix} 3& 2 &4 \\ 9& 7 & 13\\ -6&-4 &-9+x \end{bmatrix}$ has zero as an eigenvalue is ________
0 0 votes
0 0 answers
471
471 views
GATE ECE 2016 Set 2 | Question: 29
The matrix $A=\begin{bmatrix} a & 0 &3 &7 \\ 2& 5&1 &3 \\ 0& 0& 2 &4 \\ 0&0 & 0 &b \end{bmatrix}$ has $\text{det}(A) = 100$ and $\text{trace}(A) = 14$. The value of $\mid...
0 0 votes
0 0 answers
522
522 views
GATE ECE 2016 Set 1 | Question: 1
Let $M^4$= $I$,(where $I$ denotes the identity matrix) and $ M \neq I$, $M^2\neq I$ and $M^3\neq I$. Then,for any natural number $k$, $M^{-1}$ equals:$M^{4k+1}$ $M^{4...
Page: