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Recent questions and answers in Linear Algebra
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Which of the following must be an eigenvector ofA?
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May 16
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Linear Algebra
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iamHarin
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7
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#linearalgebra
#engi
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Idempotent Matrix
IF A and B are two matrices such that A.B = A, Will B always be an Identity matrix?
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May 10
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Linear Algebra
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AniPa
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6
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14
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+1
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1
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GATE2020CS18 Video Solution
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
answered
Apr 25
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Linear Algebra
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amitkhurana512
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187
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GATE2014247 Video Solution
The product of the nonzero eigenvalues of the matrix is ____ $\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}$
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Apr 18
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gate20142
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GATE201826 Video Solution
Consider a matrix P whose only eigenvectors are the multiples of $\begin{bmatrix} 1 \\ 4 \end{bmatrix}$. Consider the following statements. P does not have an inverse P has a repeated eigenvalue P cannot be diagonalized Which one of the ... III are necessarily true Only II is necessarily true Only I and II are necessarily true Only II and III are necessarily true
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Apr 18
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gate2018
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GATE2017131 Video Solution
Let $A$ be $n\times n$ real valued square symmetric matrix of rank 2 with $\sum_{i=1}^{n}\sum_{j=1}^{n}A^{2}_{ij} =$ 50. Consider the following statements. One eigenvalue must be in $\left [ 5,5 \right ]$ The eigenvalue with ... than 5 Which of the above statements about eigenvalues of $A$ is/are necessarily CORRECT? Both I and II I only II only Neither I nor II
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Apr 18
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gate20171
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GATE201713 Video Solution
Let $c_{1}.....c_{n}$ be scalars, not all zero, such that $\sum_{i=1}^{n}c_{i}a_{i}$ = 0 where $a_{i}$ are column vectors in $R^{n}$. Consider the set of linear equations $Ax = b$ ... has a unique solution at $x=J_{n}$ where $J_{n}$ denotes a $n$dimensional vector of all 1. no solution infinitely many solutions finitely many solutions
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Apr 18
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gate20171
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GATE2017130 Video Solution
Let $u$ and $v$ be two vectors in R2 whose Euclidean norms satisfy $\left \ u \right \ = 2\left \ v \right \$. What is the value of $\alpha$ such that $w = u + \alpha v$ bisects the angle between $u$ and $v$? $2$ $\frac{1}{2}$ $1$ $\frac{ 1}{2}$
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Apr 18
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gate20171
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GATE201415 Video Solution
The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a $4by4$ symmetric positive definite matrix is ___________
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Apr 18
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gate20141
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GATE2016204 Video Solution
Consider the system, each consisting of $m$ linear equations in $n$ variables. If $m < n$, then all such systems have a solution. If $m > n$, then none of these systems has a solution. If $m = n$, then there exists a system which has a solution. Which ... CORRECT? $I, II$ and $III$ are true. Only $II$ and $III$ are true. Only $III$ is true. None of them is true.
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Apr 18
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Linear Algebra
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gate20162
linearalgebra
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GATE201944 Video Solution
Consider the following matrix: $R = \begin{bmatrix} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{bmatrix}$ The absolute value of the product of Eigen values of $R$ is _______
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Apr 18
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gate2019
numericalanswers
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GATE2017252 Video Solution
If the characteristic polynomial of a 3 $\times$ 3 matrix $M$ over $\mathbb{R}$ (the set of real numbers) is $\lambda^3 – 4 \lambda^2 + a \lambda +30, \quad a \in \mathbb{R}$, and one eigenvalue of $M$ is 2, then the largest among the absolute values of the eigenvalues of $M$ is _______
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Apr 18
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gate20172
engineeringmathematics
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GATE2016105 Video Solution
Two eigenvalues of a $3 \times 3$ real matrix $P$ are $(2+\sqrt {1})$ and $3$. The determinant of $P$ is _______
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Apr 18
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gate20161
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GATE2007IT2 Video Solution
Let $A$ be the matrix $\begin{bmatrix}3 &1 \\ 1&2\end{bmatrix}$. What is the maximum value of $x^TAx$ where the maximum is taken over all $x$ that are the unit eigenvectors of $A?$ $5$ $\frac{(5 + √5)}{2}$ $3$ $\frac{(5  √5)}{2}$
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gate2007it
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GATE200426 Video Solution
The number of different $n \times n $ symmetric matrices with each element being either 0 or 1 is: (Note: $\text{power} \left(2, X\right)$ is same as $2^X$) $\text{power} \left(2, n\right)$ $\text{power} \left(2, n^2\right)$ $\text{power} \left(2,\frac{ \left(n^2+ n \right) }{2}\right)$ $\text{power} \left(2, \frac{\left(n^2  n\right)}{2}\right)$
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gate2004
linearalgebra
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GATE200727 Video Solution
Consider the set of (column) vectors defined by$X = \left \{x \in R^3 \mid x_1 + x_2 + x_3 = 0, \text{ where } x^T = \left[x_1,x_2,x_3\right]^T\right \}$ ... independent set, but it does not span $X$ and therefore is not a basis of $X$. $X$ is not a subspace of $R^3$. None of the above
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Apr 18
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Linear Algebra
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3.6k
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gate2007
linearalgebra
normal
vectorspace
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GATE201414 Video Solution
Consider the following system of equations: $3x + 2y = 1 $ $4x + 7z = 1 $ $x + y + z = 3$ $x  2y + 7z = 0$ The number of solutions for this system is ______________
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Apr 18
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Linear Algebra
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gate20141
linearalgebra
systemofequations
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GATE200725 Video Solution
Let A be a $4 \times 4$ matrix with eigen values 5,2,1,4. Which of the following is an eigen value of the matrix$\begin{bmatrix} A & I \\ I & A \end{bmatrix}$, where $I$ is the $4 \times 4$ identity matrix? $5$ $7$ $2$ $1$
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Apr 18
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Linear Algebra
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1
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gate2007
eigenvalue
linearalgebra
difficult
videosolution
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GATE200341 Video Solution
Consider the following system of linear equations ... linearly dependent. For how many values of $\alpha$, does this system of equations have infinitely many solutions? \(0\) \(1\) \(2\) \(3\)
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Apr 18
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Linear Algebra
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gate2003
linearalgebra
systemofequations
normal
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GATE2015118 Video Solution
In the LU decomposition of the matrix $\begin{bmatrix}2 & 2 \\ 4 & 9\end{bmatrix}$, if the diagonal elements of $U$ are both $1$, then the lower diagonal entry $l_{22}$ of $L$ is_________________.
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Apr 18
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Linear Algebra
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gate20151
linearalgebra
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0
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GATE2015333 Video Solution
If the following system has nontrivial solution, $px + qy + rz = 0$ $qx + ry + pz = 0$ $rx + py + qz = 0$, then which one of the following options is TRUE? $p  q + r = 0 \text{ or } p = q = r$ $p + q  r = 0 \text{ or } p = q = r$ $p + q + r = 0 \text{ or } p = q = r$ $p  q + r = 0 \text{ or } p = q = r$
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Apr 18
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gate20153
linearalgebra
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GATE2017222 Video Solution
Let $P = \begin{bmatrix}1 & 1 & 1 \\2 & 3 & 4 \\3 & 2 & 3\end{bmatrix}$ and $Q = \begin{bmatrix}1 & 2 &1 \\6 & 12 & 6 \\5 & 10 & 5\end{bmatrix}$ be two matrices. Then the rank of $ P+Q$ is ___________ .
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Linear Algebra
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gate20172
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GATE2016206 Video Solution
Suppose that the eigenvalues of matrix $A$ are $1, 2, 4$. The determinant of $\left(A^{1}\right)^{T}$ is _________.
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Apr 18
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Linear Algebra
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gate20162
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GATE19961.7 Video Solution
Let $Ax = b$ be a system of linear equations where $A$ is an $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is an $n \times1$ column vector of unknowns. Which of the following is false? The system has a solution if and only ... a unique solution. The system will have only a trivial solution when $m=n$, $b$ is the zero vector and $\text{rank}(A) =n$.
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Apr 18
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gate1996
linearalgebra
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GATE200427 Video Solution
Let $A, B, C, D$ be $n \times n$ matrices, each with nonzero determinant. If $ABCD = I$, then $B^{1}$ is $D^{1}C^{1}A^{1}$ $CDA$ $ADC$ Does not necessarily exist
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Apr 18
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gate2004
linearalgebra
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GATE2015315 Video Solution
In the given matrix $\begin{bmatrix} 1 & 1 & 2 \\ 0 & 1 & 0 \\ 1 & 2 & 1 \end{bmatrix}$ ... $\left\{a\left( \sqrt{2},0,1\right) \mid a \neq 0, a \in \mathbb{R}\right\}$
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Apr 18
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Linear Algebra
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gate20153
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GATE200476 Video Solution
In an $M \times N$ matrix all nonzero entries are covered in $a$ rows and $b$ columns. Then the maximum number of nonzero entries, such that no two are on the same row or column, is $\leq a +b$ $\leq \max(a, b)$ $\leq \min(Ma, Nb)$ $\leq \min(a, b)$
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Apr 18
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gate2004
linearalgebra
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GATE201817 Video Solution
Consider a matrix $A= uv^T$ where $u=\begin{pmatrix}1 \\ 2 \end{pmatrix} , v = \begin{pmatrix}1 \\1 \end{pmatrix}$. Note that $v^T$ denotes the transpose of $v$. The largest eigenvalue of $A$ is ____
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Apr 18
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gate2018
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GATE201211 Video Solution
Let A be the $ 2 × 2 $ matrix with elements $a_{11} = a_{12} = a_{21} = +1 $ and $ a_{22} = −1 $ . Then the eigenvalues of the matrix $A^{19}$ are $1024$ and $−1024$ $1024\sqrt{2}$ and $−1024 \sqrt{2}$ $4 \sqrt{2}$ and $−4 \sqrt{2}$ $512 \sqrt{2}$ and $−512 \sqrt{2}$
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gate2012
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GATE20083 Video Solution
The following system of equations $x_1 + x_2 + 2x_3 = 1$ $x_1 + 2x_2 + 3x_3 = 2$ $x_1 + 4x_2 + αx_3 = 4$ has a unique solution. The only possible value(s) for $α$ is/are $0$ either $0$ or $1$ one of $0, 1$, or $1$ any real number
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gate2008
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GATE201424 Video Solution
If the matrix $A$ is such that $A= \begin{bmatrix} 2\\ −4\\7\end{bmatrix}\begin{bmatrix}1& 9& 5\end{bmatrix}$ then the determinant of $A$ is equal to ______.
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gate20142
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GATE200623 Video Solution
$F$ is an $n\times n$ real matrix. $b$ is an $n\times 1$ real vector. Suppose there are two $n\times 1$ vectors, $u$ and $v$ such that, $u ≠ v$ and $Fu = b, Fv = b$. Which one of the following statements is false? Determinant of $F$ is zero. There are an infinite number of solutions to $Fx = b$ There is an $x≠0$ such that $Fx = 0$ $F$ must have two identical rows
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gate2006
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GATE199301.1 Video Solution
For the below question, one or more of the alternatives are correct. Write the code letter$(s)$ $a$, $b$, $c$, $d$ corresponding to the correct alternative$(s) $ in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative ... $(0,0,\alpha)$ $(\alpha,0,0)$ $(0,0,1)$ $(0,\alpha,0)$
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Linear Algebra
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gate1993
eigenvalue
linearalgebra
easy
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GATE201434 Video Solution
Which one of the following statements is TRUE about every $n \times n$ matrix with only real eigenvalues? If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative. ... eigenvalues are positive. If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
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gate20143
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GATE20199 Video Solution
Let $X$ be a square matrix. Consider the following two statements on $X$. $X$ is invertible Determinant of $X$ is nonzero Which one of the following is TRUE? I implies II; II does not imply I II implies I; I does not imply II I does not imply II; II does not imply I I and II are equivalent statements
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gate2019
engineeringmathematics
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GATE201435 Video Solution
If $V_1$ and $V_2$ are $4$dimensional subspaces of a $6$dimensional vector space $V$, then the smallest possible dimension of $V_1 \cap V_2$ is _____.
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gate20143
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GATE20133 Video Solution
Which one of the following does NOT equal $\begin{vmatrix} 1 & x & x^{2}\\ 1& y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} \quad ?$ $\begin{vmatrix} 1& x(x+1)& x+1\\ 1& y(y+1) & y+1\\ 1& z(z+1) & z+1 \end{vmatrix}$ ...
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gate2013
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GATE2004IT32 Video Solution
Let $A$ be an $n \times n$ ...
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GATE2008IT29 Video Solution
If $M$ is a square matrix with a zero determinant, which of the following assertion (s) is (are) correct? S1: Each row of $M$ can be represented as a linear combination of the other rows S2: Each column of $M$ can be represented as a linear combination of the other columns ... solution S4: $M$ has an inverse $S3$ and $S2$ $S1$ and $S4$ $S1$ and $S3$ $S1, S2$ and $S3$
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GATE2005IT3 Video Solution
The determinant of the matrix given below is $\begin{bmatrix} 0 &1 &0 &2 \\ 1& 1& 1& 3\\ 0&0 &0 & 1\\ 1& 2& 0& 1 \end{bmatrix}$ $1$ $0$ $1$ $2$
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