Recent questions and answers in Linear Algebra - GATE CSE Doubts

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Linear Dependency
I have a confusion, let's say we have three vectors a,b & c in 2-D space. a & b are linearly independent but c is dependent on a & b. Can't we represent whole 2-D space using a,b & c as (any vector V) V = x*a + y*b +0*c and hence a,b & c form basis. Then how can we say that basis are set of linearly independent vectors. Please tell me where I am going wrong.

asked 5 days ago in Linear Algebra by Codered03 (5 points) | 5 views

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TIFR-GS2021 Question
Let $M$ be a $n\times m$ real matrix. Consider the following: Let $k_1$ be the smallest number such that $M$ can be factorized as $A.B$, where $A$ is an $n\times k_1$ matrix and $B$ is a $k_1\times m$ matrix. Let $k_2$ ... $k_2 = k_3 < k_1$ (D) $k_1 = k_2 = k_3$ (E) No general relationship exists among $k_1$, $k_2$ and $k_3$

asked Mar 24 in Linear Algebra by zxy123 (3.6k points) | 6 views

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Gate overflow book
https://gateoverflow.in/204100/gate2018-26. The above is the link for the question. I am not able to understand the explaination for statement I i.e. the matrix may have an inverse. Can someone provide a proof or a counter example for the specific situation when the eigenvectors are multiples of a single vector the matrix may or may not have an inverse?

asked Jan 19 in Linear Algebra by hadarsh (5 points) | 14 views

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Testbook test series linear algebra
Let M be a real 4X4 matrix. Consider the following: s1: M has 4 linearly independent eigenvectors s2: M has 4 distinct eigenvalues s3: M is non singular (invertible) Which of the following is true? s1 implies s2 s1 implies s3 s2 implies s1 s3 implies s2 The answer given was s2 implies s1 which is fine. But why not also s1 implies s2?

asked Dec 10, 2020 in Linear Algebra by Dheera (-14 points) | 41 views

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No. of non-zero eigenvalues <= Rank of the matrix
Does anyone knows the proof of this concept, I am not able to find it. Reference question – https://gateoverflow.in/2013/gate2014-2-47

asked Nov 27, 2020 in Linear Algebra by smn98 (5 points) | 38 views

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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?

answered Aug 16, 2020 in Linear Algebra by Arkaprava (801 points) | 36 views

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Which of the following must be an eigenvector ofA?

answered Aug 15, 2020 in Linear Algebra by Arkaprava (801 points) | 22 views

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GATE2020-CS-18 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, 2020 in Linear Algebra by amitkhurana512 (47 points) | 33 views

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GATE2014-2-47 Video Solution
The product of the non-zero eigenvalues of the matrix is ____ $\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 20 views

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GATE2018-26 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

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 14 views

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GATE2017-1-31 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

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 8 views

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GATE2017-1-3 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

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE2017-1-30 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 $v$? $2$ $\frac{1}{2}$ $1$ $\frac{ -1}{2}$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 13 views

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GATE2014-1-5 Video Solution
The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a $4-by-4$ symmetric positive definite matrix is ___________

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 10 views

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GATE2016-2-04 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.

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 13 views

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GATE2019-44 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 _______

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 11 views

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GATE2017-2-52 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 _______

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 24 views

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GATE2016-1-05 Video Solution
Two eigenvalues of a $3 \times 3$ real matrix $P$ are $(2+\sqrt {-1})$ and $3$. The determinant of $P$ is _______

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 11 views

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GATE2007-IT-2 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 $A?$ $5$ $\frac{(5 + √5)}{2}$ $3$ $\frac{(5 - √5)}{2}$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 4 views

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GATE2004-26 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 $2^X$) $\text{power} \left(2, n\right)$ $\text{power} \left(2, n^2\right)$ $\text{power} \left(2,\frac{ \left(n^2+ n \right) }{2}\right)$ $\text{power} \left(2, \frac{\left(n^2 - n\right)}{2}\right)$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 10 views

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GATE2007-27 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

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE2014-1-4 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 ______________

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE2007-25 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? $-5$ $-7$ $2$ $1$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 8 views

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GATE2003-41 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? \(0\) \(1\) \(2\) \(3\)

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE2015-1-1‏8 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_________________.

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 9 views

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GATE2015-3-33 Video Solution
If the following system has non-trivial solution, $px + qy + rz = 0$ $qx + ry + pz = 0$ $rx + py + qz = 0$, then which one of the following options is TRUE? $p - q + r = 0 \text{ or } p = q = -r$ $p + q - r = 0 \text{ or } p = -q = r$ $p + q + r = 0 \text{ or } p = q = r$ $p - q + r = 0 \text{ or } p = -q = -r$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE2017-2-22 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 ___________ .

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 6 views

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GATE2016-2-06 Video Solution
Suppose that the eigenvalues of matrix $A$ are $1, 2, 4$. The determinant of $\left(A^{-1}\right)^{T}$ is _________.

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE1996-1.7 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 $\text{rank}(A) =n$.

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE2004-27 Video Solution
Let $A, B, C, D$ be $n \times n$ matrices, each with non-zero determinant. If $ABCD = I$, then $B^{-1}$ is $D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE2015-3-15 Video Solution
In the given matrix $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 1 & 0 \\ 1 & 2 & 1 \end{bmatrix}$ ... $\left\{a\left(- \sqrt{2},0,1\right) \mid a \neq 0, a \in \mathbb{R}\right\}$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE2004-76 Video Solution
In an $M \times N$ matrix all non-zero entries are covered in $a$ rows and $b$ columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is $\leq a +b$ $\leq \max(a, b)$ $\leq \min(M-a, N-b)$ $\leq \min(a, b)$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 9 views

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GATE2018-17 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 ____

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 6 views

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GATE2012-11 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 $−512 \sqrt{2}$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 6 views

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GATE2008-3 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

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 9 views

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GATE2014-2-4 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 ______.

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 9 views

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GATE2006-23 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

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 7 views

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GATE1993-01.1 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 ... $(0,0,\alpha)$ $(\alpha,0,0)$ $(0,0,1)$ $(0,\alpha,0)$

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 16 views

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GATE2014-3-4 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.

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 17 views

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GATE2019-9 Video Solution
Let $X$ be a square matrix. Consider the following two statements on $X$. $X$ is invertible Determinant of $X$ is non-zero 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

asked Apr 18, 2020 in Linear Algebra by admin (573 points) | 8 views

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