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Recent questions and answers in Linear Algebra
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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 we conclude? $\mathbf{A}$ is invertible $\mathbf{A}^{T}=\mathbf{A}$ $\mathbf{A}^{2}=\mathbf{A}$ Only (i) Only (ii) Only (iii) Only (i) and (ii) None of the above
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tifr2015
linear-algebra
matrices
1
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2
TIFR ECE 2015 | Question: 7
Let $A$ be an $8 \times 8$ matrix of the form \[ \left[\begin{array}{cccc} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \ldots & 2 \end{array}\ ... $\operatorname{det}(A)=9$ $\operatorname{det}(A)=18$ $\operatorname{det}(A)=14$ $\operatorname{det}(A)=27$ None of the above
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tifr2015
linear-algebra
determinant
1
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0
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3
TIFR ECE 2015 | Question: 13
Let \[ A=\left(\begin{array}{ccc} 1 & 1+\varepsilon & 1 \\ 1+\varepsilon & 1 & 1+\varepsilon \\ 1 & 1+\varepsilon & 1 \end{array}\right) \] Then for $\varepsilon=10^{-6}, A$ has only negative eigenvalues only non-zero eigenvalues only positive eigenvalues one negative and one positive eigenvalue None of the above
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Linear Algebra
Dec 15, 2022
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tifr2015
linear-algebra
eigen-values
1
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0
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4
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 $a_{1}>a_{2}>a_{3}$ $a_{1}<a_{2}<a_{3}$ $a_{1}=3, a_{2}=2, a_{3}=4$ All of the above None of the above
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Dec 14, 2022
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tifr2014
linear-algebra
matrices
1
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5
TIFR ECE 2014 | Question: 7
Let $A$ be an $n \times n$ real matrix. It is known that there are two distinct $n$-dimensional real column vectors $v_{1}, v_{2}$ such that $A v_{1}=A v_{2}$. Which of the following can we conclude about $A?$ All eigenvalues of $A$ are non-negative. $A$ is not full rank. $A$ is not the zero matrix. $\operatorname{det}(A) \neq 0$. None of the above.
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Dec 14, 2022
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tifr2014
linear-algebra
eigen-values
1
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6
TIFR ECE 2013 | Question: 10
Consider the following series of square matrices: \[ \begin{array}{l} H_{1}=[1], \\ H_{2}=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right], \end{array} \] and for $k=2,3, \ldots$, the $2^{k} \times 2^{k}$ matrix $H_{2^{k}}$ is recursively defined as \[ H_{2^{k}}=\ ... is $H_{2^{k}} H_{2^{k}}^{T}?)$ $0$ $2^{k}$ $2^{k / 2}$ $2^{k 2^{k-1}}$ $2^{k 2^{k}}$
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Dec 12, 2022
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tifr2013
linear-algebra
determinant
1
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0
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7
TIFR ECE 2013 | Question: 11
Two matrices $A$ and $B$ are called similar if there exists another matrix $S$ such that $S^{-1} A S=B$. Consider the statements: If $A$ and $B$ are similar then they have identical rank. If $A$ and $B$ ... Both $\text{I}$ and $\text{II}$ but not $\text{III}$. All of $\text{I}, \text{II}$ and $\text{III}$.
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Dec 12, 2022
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tifr2013
linear-algebra
rank-of-matrix
1
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0
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8
TIFR ECE 2013 | Question: 12
Let $A$ be a Hermitian matrix and let $I$ be the Identity matrix with same dimensions as $A$. Then for a scalar $\alpha>0, A+\alpha I$ has the same eigenvalues as of $A$ but different eigenvectors the same eigenvalues and eigenvectors as of ... those of $A$ and same eigenvectors as of $A$ eigenvalues and eigenvectors with no relation to those of $A$ None of the above
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Dec 12, 2022
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tifr2013
linear-algebra
eigen-values
1
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9
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: $x^{\dagger} A^{2} x$ is always non-negative $x^{\dagger} A x$ ... $A=A^{\dagger}$ then $x^{\dagger} A y$ is complex for some vector $y$ with same dimensions as $x$
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Dec 12, 2022
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tifr2013
linear-algebra
matrices
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10
TIFR ECE 2012 | Question: 16
Let $P$ be a $n \times n$ matrix such that $P^{k}=\mathbf{0}$, for some $k \in \mathbb{N}$ and where $\mathbf{0}$ is an all zeros matrix. Then at least how many eigenvalues of $P$ are zero $1$ $n-1$ $n$ $0$ None of the above
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Dec 8, 2022
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tifr2012
linear-algebra
eigen-values
1
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11
TIFR ECE 2012 | Question: 17
Let $A=U \Lambda U^{\dagger}$ be a $n \times n$ matrix, where $U U^{\dagger}=I$. Which of the following statements is TRUE. The matrix $I+A$ has non-negative eigen values The matrix $I+A$ is symmetic $\operatorname{det}(I+A)=\operatorname{det}(I+\Lambda)$ $(a)$ and $(c)$ $(b)$ and $(c)$ $(a), (b)$ and $(c)$
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Dec 8, 2022
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tifr2012
linear-algebra
eigen-values
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12
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 right side. If the transformation is of the form \[ \left[\begin{array}{l} u \\ v \end{array}\right]=\mathbf{A}\ ... \] $A_{1}$ only $A_{2}$ only $A_{1}$ or $A_{2}$ $A_{1}$ or $A_{3}$ $A_{2}$ or $A_{3}$
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Dec 8, 2022
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tifr2012
linear-algebra
matrices
1
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0
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13
TIFR ECE 2012 | Question: 19
$X$ and $Y$ are two $3$ by $3$ matrices. If \[ X Y=\left(\begin{array}{rrr} 1 & 3 & -2 \\ -4 & 2 & 5 \\ 2 & -8 & -1 \end{array}\right) \] then $X$ has rank $2$ at least one of $X, Y$ is not invertible $X$ can't be an invertible matrix $X$ and $Y$ could both be invertible. None of the above
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Dec 8, 2022
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tifr2012
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14
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 $B=e^{2} A / 2$ $B=\left(\begin{array}{cc}1+e & e \\e & 1+e\end{array}\right)$ ... $B=\left(\begin{array}{cc}1+e^{2} & e^{2} \\e^{2} & 1+e^{2}\end{array}\right)$ None of the above
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tifr2012
linear-algebra
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15
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 the following is true? $F H F^{H}$ is circulant, where $F^{H}$ is the inverse $\text{DFT}$ matrix. $F H F^{H}$ is tridiagonal $F H F^{H}$ is diagonal $F H F^{H}$ has real entries None of the above
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Dec 1, 2022
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tifr2010
linear-algebra
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1
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16
TIFR ECE 2010 | Question: 11
Consider \[ \text{F}=\frac{1}{2}\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & -1 \end{array}\right], \quad x=\left[\begin{array}{l} 2.1 \\ 1.2 \\ ... 2 \\ -1 \end{array}\right] \] The inner product between $\text{F}x$ and $\text{F}y$ is $0$ $1$ $-1$ $-1.2$ None of the above
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tifr2010
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17
TIFR ECE 2022 | Question: 8
Let $a, b, c$ be real numbers such that the following system of equations has a solution \[\begin{aligned} x+2 y+3 z &=a & & (1)\\ 8 x+10 y+12 z &=b & & (2)\\ 7 x+8 y+9 z &=c-1 & & (3) \end{aligned}\] Let $A$ be a ... 1 & 0 \\ -1 & 0 & 1 \end{array}\right]\] What is the value of $\operatorname{det}(A)$? $1$ $2$ $3$ $4$ $5$
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Nov 30, 2022
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tifrece2022
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18
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 following statements: $\operatorname{det}(\mathbf{P})$ is either $+1$ ... $1,3$ are correct Only statements $2, 3$ are correct All statements $1, 2,$ and $3$ are correct
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tifrece2022
linear-algebra
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19
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 zero All eigenvalues of $A$ are zero All diagonal entries of $A$ are zero ... $2$ alone is correct Only statement $3$ is correct Only statements $1$ and $2$ are correct Only statements $2$ and $3$ are correct
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tifrece2020
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20
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 following statements: $\operatorname{det}(x I-A)=\operatorname{det}(x I-B)$ ... statement $2$ is correct Only statements $1$ and $2$ are correct All Statements $1, 2$ and $3$ are correct None of the above
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tifrece2020
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21
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 always $\text{TRUE}?$ $A B$ has full rank $A-B$ ... an eigenvalue of $A B$ $A+B$ has full rank At least one of $\lambda_{A}$ or $\lambda_{B}$ is an eigenvalue of $A B$
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tifrece2019
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22
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 the following statements. At least one eigenvalue of $A$ is zero. $A$ ... $\text{(i)}$ Only $\text{(ii)}$ Only $\text{(iii)}$ Only $\text{(iv)}$ All of $\text{(i) - (iv)}$
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tifrece2018
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23
TIFR ECE 2017 | Question: 7
A circulant matrix is a square matrix whose each row is the preceding row rotated to the right by one element, e.g., the following is a $3 \times 3$ circulant matrix. \[\left(\begin{array}{lll} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \ ... $j=\sqrt{-1}$ A vector whose $k$-th element is $\sinh \left(\frac{2 \pi k}{n}\right)$ None of the above
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tifrece2017
linear-algebra
eigen-values
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24
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 $A v_{1}=A v_{2}$. $A$ can be similar to the identity matrix, when $A$ is not the identity matrix. Recall that ... $\text{(ii)}$ Only $\text{(iii)}$ $\text{(i), (ii),}$ and $\text{(iii)}$ None of the above
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tifrece2017
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25
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. $\operatorname{rank}\left(A A^{T}\right)=\operatorname{rank}\left(A^{T} A\right)$ ... Which of the above statements is true for all such $A?$ Only (i) Only (ii) Only (iii) (i) and (iii) None of them
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26
TIFR ECE 2016 | Question: 14
Consider matrices $A \in \mathbb{R}^{n \times m}, B \in \mathbb{R}^{m \times m}$, and $C \in \mathbb{R}^{m \times n}$. Let $r=\operatorname{rank}(A B C)$. Which of the following must be true? $\min (m, n) \leq r \leq \max (m, n)$ ... $\min (m, n) \leq r \leq \max (\operatorname{rank}(A), \operatorname{rank}(B), \operatorname{rank}(C))$ None of the above
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27
TIFR ECE 2016 | Question: 15
What is \[ \max _{x, y}\left[\begin{array}{ll} x & y \end{array}\right]\left[\begin{array}{cc} 3 & \sqrt{2} \\ \sqrt{2} & 2 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right] \] subject to \[ x^{2}+y^{2}=1 ? \] $1$ $\sqrt{2}$ $2$ $3$ $4$
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28
GATE ECE 2010 | Question: 1
The eigenvalues of a skew-symmetric matrix are always zero always pure imaginary either zero or pure imaginary always real
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29
GATE ECE 2011 | Question: 35
The system of equations $ \begin{aligned} &x+y+z=6 \\ &x+4 y+6 z=20 \\ &x+4 y+\lambda z=\mu \end{aligned} $ has NO solution for values of $\lambda$ and $\mu$ given by $\lambda=6, \mu=20$ $\lambda=6, \mu \neq 20$ $\lambda \neq 6, \mu=20$ $\lambda \neq 6, \mu \neq 20$
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30
GATE ECE 2021 | Question: 36
A real $2\times2$ non-singular matrix $A$ with repeated eigenvalue is given as $A=\begin{bmatrix} x & -3.0\\ 3.0 & 4.0 \end{bmatrix}$ where $x$ is a real positive number. The value of $x$ (rounded off to one decimal place) is ________________
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31
GATE ECE 2020 | Question: 26
Consider the following system of linear equations. $\begin{array}{llll} x_{1}+2x_{2}=b_{1} ; & 2x_{1}+4x_{2}=b_{2}; & 3x_{1}+7x_{2}=b_{3} ; & 3x_{1}+9x_{2}=b_{4} \end{array}$ Which one of the following conditions ensures that a solution exists for the above system? ... $b_{2}=2b_{1}$ and $3b_{1}-6b_{3}+b_{4}=0$ $b_{3}=2b_{1}$ and $3b_{1}-6b_{3}+b_{4}=0$
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32
GATE ECE 2013 | Question: 19
The minimum eigenvalue of the following matrix is $\begin{bmatrix} 3& 5& 2\\5 &12 &7 \\2 &7 & 5\end{bmatrix}$ $0$ $1$ $2$ $3$
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33
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 ____________.
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34
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 $\textbf{A}$ are $(\sigma + j\omega)$ and $(\sigma - j\omega)$, then $x$ is equal to $+j\omega$ $-j\omega$ $+\omega$ $-\omega$
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35
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 ________
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36
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 $\mid a-b \mid$ is ________
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37
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: $M^{4k+1}$ $M^{4k+2}$ $M^{4k+3}$ $M^{4k}$
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38
GATE ECE 2016 Set 1 | Question: 27
A sequence $x[n]$ is specified as $\begin{bmatrix}x[n] \\x[n – 1]\end{bmatrix}=\begin{bmatrix}1&1\\1&0\end{bmatrix}^n\begin{bmatrix}1\\0\end{bmatrix}$,for $n \geq 2$. The initial conditions are $x[0] = 1$, $x[1] = 1$, and $x[n] = 0$ for $n < 0$. The value of $x[12]$ is _________
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GATE ECE 2015 Set 3 | Question: 1
For $A = \begin{bmatrix} 1 &\tan x \\ -\tan x &1 \end{bmatrix},$ the determinant of $A^{T}A^{-1}$ is $\sec^{2}x$ $\cos 4x$ $1$ $0$
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40
GATE ECE 2015 Set 2 | Question: 2
The value of $x$ for which all the eigen-values of the matrix given below are real is $\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &-10 \end{bmatrix}$ $5+j$ $5-j$ $1-5j$ $1+5j$
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