Recent questions and answers in Linear Algebra

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Eigen value question from Linear algebra and its applications

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

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1 answer 20 views

Linear Algebra and Its Applications Fourth Edition Gilbert Strang, chapter 1, problem set 1.2, Qn. 10

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

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0 answers 21 views

can we say that two parallel vectors (even with different origins) are linearly dependent and hence can't form basis?

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

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0 answers 14 views

Matrices, mathematics
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)

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

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0 answers 22 views

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.

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

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0 answers 14 views

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 $k_3$

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

GATE CSE 2021 Set 2 | Question: 24 | Video Solution

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

GATE CSE 2021 Set 1 | Question: 52 | Video Solution

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

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0 answers 16 views

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?

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

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?

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

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

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

Idempotent Matrix
IF A and B are two matrices such that A.B = A, Will B always be an Identity 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 Arkaprava 867 points 44 views

0 votes

1 answer 25 views

Which of the following must be an eigenvector ofA?

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

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 _______.

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 amitkhurana512 173 points 39 views

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0 answers 23 views

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

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

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0 answers 21 views

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

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

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0 answers 11 views

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

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

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0 answers 12 views

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

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

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0 answers 17 views

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

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

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0 answers 14 views

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 ___________

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

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0 answers 19 views

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.

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

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0 answers 16 views

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 _______

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

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0 answers 33 views

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 _______

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

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0 answers 14 views

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 _______

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

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0 answers 9 views

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

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

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0 answers 13 views

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

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

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0 answers 11 views

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

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

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 ______________

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

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0 answers 12 views

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? $-5$ $-7$ $2$ $1$

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

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? $0$ $1$ $2$ $3$

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

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_________________.

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

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? $p - q + r = 0 \text{ or } p = q = -r$ $p + q - r = 0 \text{ or } p = -q = r$ $p + q + r = 0 \text{ or } p = q = r$ $p - q + r = 0 \text{ or } p = -q = -r$

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

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0 answers 8 views

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 ___________ .

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

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 _________.

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

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

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

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

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

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

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

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 $\leq a +b$ $\leq \max(a, b)$ $\leq \min(M-a, N-b)$ $\leq \min(a, b)$

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

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 ____

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

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 $−512 \sqrt{2}$

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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