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Recent questions and answers in Linear Algebra

0 votes
1 answer 17 views
Find the eigenvalues and the eigenvectors of these two matrices: A = \begin{bmatrix}1&4\\2&3\end{bmatrix} and A+I = \begin{bmatrix}2&4\\2&4\end{bmatrix} A+I has the eigenvectors _____ as A. Its eigenvalues are ______ by 1. I have computed the eigen values please help me in determining a relation between them. Is there any formula or theorem I’m missing out on?
answered Jul 11 in Linear Algebra asqwer 633 points 17 views
0 votes
1 answer 20 views
Under what condition on y1, y2, y3 do the points (0;y1), (1;y2), (2;y3) lie on a straight line?
answered May 15 in Linear Algebra aryavart 73 points 20 views
0 votes
0 answers 21 views
Since both the vectors have same direction so the they must not be basis as they are linearly dependent but this link says something else, Please explain Math Stack Exchange – https://math.stackexchange.com/questions/1997469/basis-for-the-set-of-parallel-vectors#:~:text=1%20Answer&text=Yes.,exactly%20the%20span%20of%20v.
asked May 2 in Linear Algebra Codered03 5 points 21 views
0 votes
0 answers 14 views
The number of different n × n symmetric matrices with each element being either 0 or 1 is: (Note: power (2, x) is same as 2x)
asked May 2 in Linear Algebra Aarvin 5 points 14 views
0 votes
0 answers 22 views
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 Apr 14 in Linear Algebra Codered03 5 points 22 views
0 votes
0 answers 14 views
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 zxy123 3.6k points 14 views
5 votes
3 answers 1K views
Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $\text{Px=0}$ is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$. The rank of $P$ is __________
asked Feb 18 in Linear Algebra Arjun 1.5k points 1K views
1 vote
2 answers 832 views
Consider the following matrix.$\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}$The largest eigenvalue of the above matrix is __________.
asked Feb 18 in Linear Algebra Arjun 1.5k points 832 views
0 votes
0 answers 16 views
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 hadarsh 5 points 16 views
1 vote
0 answers 44 views
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 Dheera -14 points 44 views
0 votes
0 answers 61 views
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 smn98 5 points 61 views
0 votes
1 answer 44 views
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 Arkaprava 867 points 44 views
0 votes
1 answer 25 views
Let A be a 2×2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvector of A? A.[0 1] B.[1 0] c.[1 1]
answered Aug 15, 2020 in Linear Algebra Arkaprava 867 points 25 views
1 vote
1 answer 39 views
0 votes
0 answers 23 views
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 admin 573 points 23 views
0 votes
0 answers 21 views
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 following options is correct ... I and 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 admin 573 points 21 views
0 votes
0 answers 11 views
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 the largest ... strictly greater 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 admin 573 points 11 views
0 votes
0 answers 12 views
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 $Ax = b$ where $A=\left [ a_{1}.....a_{n} \right ]$ ... set of equations 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 admin 573 points 12 views
0 votes
0 answers 17 views
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 admin 573 points 17 views
0 votes
0 answers 14 views
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 admin 573 points 14 views
0 votes
0 answers 19 views
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 one of the following is 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 admin 573 points 19 views
0 votes
0 answers 16 views
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 admin 573 points 16 views
0 votes
0 answers 33 views
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 admin 573 points 33 views
0 votes
0 answers 14 views
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 admin 573 points 14 views
0 votes
0 answers 9 views
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 admin 573 points 9 views
0 votes
0 answers 13 views
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 admin 573 points 13 views
0 votes
0 answers 11 views
Consider the set of (column) vectors defined by$X = \left \{x \in R^3 \mid x_1 + x_2 + x_3 = 0, \text{ where } x^T = \left[x_1,x_2,x_3\right]^T\right \}$.Which of the following is TRUE? $\left\{\left[1,-1,0\right]^T,\left[1,0,-1\right]^T\right\}$ is a ... is a linearly 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 admin 573 points 11 views
0 votes
0 answers 13 views
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 admin 573 points 13 views
0 votes
0 answers 12 views
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 admin 573 points 12 views
0 votes
0 answers 13 views
Consider the following system of linear equations ... matrix are 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 admin 573 points 13 views
0 votes
0 answers 15 views
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 admin 573 points 15 views
0 votes
0 answers 12 views
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 admin 573 points 12 views
0 votes
0 answers 8 views
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 admin 573 points 8 views
0 votes
0 answers 10 views
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 admin 573 points 10 views
0 votes
0 answers 11 views
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 if, both $A$ ... system has 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 admin 573 points 11 views
0 votes
0 answers 10 views
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 admin 573 points 10 views
0 votes
0 answers 11 views
In the given matrix $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 1 & 0 \\ 1 & 2 & 1 \end{bmatrix}$ , one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are $\left\{a\left(4,2,1\right) \mid a \neq 0, a \in \mathbb{R}\right\}$ ... $\left\{a\left(- \sqrt{2},0,1\right) \mid a \neq 0, a \in \mathbb{R}\right\}$
asked Apr 18, 2020 in Linear Algebra admin 573 points 11 views
0 votes
0 answers 12 views
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 admin 573 points 12 views
0 votes
0 answers 9 views
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 admin 573 points 9 views
0 votes
0 answers 10 views
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 admin 573 points 10 views
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