Recent questions in Linear Algebra

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NIELIT 2016 MAR Scientist B - Section B: 4
What is the determinant of the matrix $\begin{bmatrix}5&3&2\\1&2&6\\3&5&10\end{bmatrix}$ $-76$ $-28$ $+28$ $+72$

asked Mar 31 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 47 views

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NIELIT 2016 MAR Scientist B - Section B: 6
The system of simultaneous equations $x+2y+z=6\\2x+y+2z=6\\x+y+z=5\\\text{has}$ unique solution. infinite number of solutions. no solution. exactly two solutions.

asked Mar 31 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 40 views

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NIELIT 2016 MAR Scientist B - Section B: 12
If $A$ and $B$ are square matrices of size $n\times n$, then which of the following statements is not true? $det(AB)=det(A)det(B)$ $det(kA)=k^ndet(A)$ $det(A+B)=det(A)+det(B)$ $det(A^T)=1/det(A^{-1})$

asked Mar 31 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 29 views

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4

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NIELIT 2016 DEC Scientist B (IT) - Section B: 26
Two eigenvalues of a $3\times3$ real matrix $P$ are $(2+ \sqrt-1)$ and $3$. The determinant of $P$ is ________. $0$ $1$ $15$ $-1$

asked Mar 31 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 19 views

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5

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NIELIT 2016 DEC Scientist B (CS) - Section B: 21
Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is: $D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist.

asked Mar 31 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 26 views

+1 vote

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UGCNET-Dec2004-II: 1
AVA=A is called : Identity law De Morgan s law Idempotent law Complement law

asked Mar 26 in Linear Algebra by jothee Veteran (109k points) 32 views

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Introduction to Linear Algebra 4th edition Problem Set 1.1
How many corner does a cube have in 4 dimensions? How many 3D faces? Now by observation we can tell that, an n-dimensional cube has $2^n$ corners. 1D cube which is a line have $2^1$ corners 2D cube which is a square have $2^2$ corners ... but this is the question i'm not able to answer. How every N-cube have $|2n|$ cubes of dimension (N-1)?

asked Feb 26 in Linear Algebra by Mk Utkarsh Boss (37k points) 104 views

+2 votes

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GATE2020-CS-18
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.

asked Feb 12 in Linear Algebra by Arjun Veteran (436k points) 1.1k views

+3 votes

3 answers

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GATE2020-CS-27
Let $A$ and $B$ be two $n \times n$ matrices over real numbers. Let rank($M$) and $\text{det}(M)$ denote the rank and determinant of a matrix $M$, respectively. Consider the following statements. $\text{rank}(AB) = \text{rank }(A) \text{rank }(B)$ $\text{det}(AB) ... + \text{det} (B)$ Which of the above statements are TRUE? I and II only I and IV only II and III only III and IV only

asked Feb 12 in Linear Algebra by Arjun Veteran (436k points) 915 views

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TIFR2020-A-12
The hour needle of a clock is malfunctioning and travels in the anti-clockwise direction, i.e., opposite to the usual direction, at the same speed it would have if it was working correctly. The minute needle is working correctly. Suppose the two needles show the correct time at $12$ noon ... dfrac{10}{11}$ hour $\dfrac{11}{12}$ hour $\dfrac{12}{13}$ hour $\dfrac{19}{22}$ hour One hour

asked Feb 11 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 54 views

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TIFR2020-A-5
Let $A$ be am $n\times n$ invertible matrix with real entries whose column sums are all equal to $1$. Consider the following statements: Every column in the matrix $A^{2}$ sums to $2$ Every column in the matrix $A^{3}$ sums to $3$ Every column in the matrix $A^{-1}$ sums to $1$ ... (3)$ is correct but not statements $(1)$ or $(2)$ all the $3$ statements $(1),(2),$ and $(3)$ are correct

asked Feb 10 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 77 views

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TIFR2020-A-3
Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $$S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f'_{a}(w) = 0\},$$ where the polynomial function $f_{a}(x)$ is defined as $f_{a}(x):=\ ... is a $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$ $S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1}$ None of the other options

asked Feb 10 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 37 views

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1 answer

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TIFR2020-A-2
Let $M$ be a real $n\times n$ matrix such that for$ every$ non-zero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x> 0.$ Then Such an $M$ cannot exist Such $Ms$ exist and their rank is always $n$ Such $Ms$ exist, but their eigenvalues are always real No eigenvalue of any such $M$ can be real None of the above

asked Feb 10 in Linear Algebra by Lakshman Patel RJIT Veteran (63.1k points) 56 views

+1 vote

3 answers

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ISI2014-DCG-8
If $M$ is a $3 \times 3$ matrix such that $\begin{bmatrix} 0 & 1 & 2 \end{bmatrix}M=\begin{bmatrix}1 & 0 & 0 \end{bmatrix}$ and $\begin{bmatrix}3 & 4 & 5 \end{bmatrix} M = \begin{bmatrix}0 & 1 & 0 \end{bmatrix}$ then $\begin{ ... & 1 \end{bmatrix}$ $\begin{bmatrix} -1 & 2 & 0 \end{bmatrix}$ $\begin{bmatrix} 9 & 10 & 8 \end{bmatrix}$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 175 views

+2 votes

1 answer

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ISI2014-DCG-9
The values of $\eta$ for which the following system of equations $$\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$$ has a solution are $\eta=1, -2$ $\eta=-1, -2$ $\eta=3, -3$ $\eta=1, 2$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 113 views

+1 vote

1 answer

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ISI2014-DCG-25
The determinant $\begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix}$ equals $\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ $2\begin{vmatrix} a & ... & z \end{vmatrix}$ $3\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ None of these

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 88 views

+1 vote

1 answer

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ISI2014-DCG-38
Suppose that $A$ is a $3 \times 3$ real matrix such that for each $u=(u_1, u_2, u_3)’ \in \mathbb{R}^3, \: u’Au=0$ where $u’$ stands for the transpose of $u$. Then which one of the following is true? $A’=-A$ $A’=A$ $AA’=I$ None of these

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 100 views

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1 answer

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ISI2014-DCG-64
The value of $\lambda$ such that the system of equation $$\begin{array}{} 2x & – & y & + & 2z & = & 2 \\ x & – & 2y & + & z & = & -4 \\ x & + & y & + & \lambda z & = & 4 \end{array}$$ has no solution is $3$ $1$ $0$ $-3$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 92 views

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ISI2014-DCG-70
For the matrices $A = \begin{pmatrix} a & a \\ 0 & a \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, $(B^{-1}AB)^3$ is equal to $\begin{pmatrix} a^3 & a^3 \\ 0 & a^3 \end{pmatrix}$ $\begin{pmatrix} a^3 & ... $\begin{pmatrix} a^3 & 0 \\ 3a^3 & a^3 \end{pmatrix}$ $\begin{pmatrix} a^3 & 0 \\ -3a^3 & a^3 \end{pmatrix}$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 69 views

+1 vote

1 answer

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ISI2015-MMA-37
Let $a$ be a non-zero real number. Define $$f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$$ for $x \in \mathbb{R}$. Then, the number of distinct real roots of $f(x) =0$ is $1$ $2$ $3$ $4$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 96 views

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ISI2015-MMA-38
A real $2 \times 2$ matrix $M$ such that $$M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1- \varepsilon \end{pmatrix}$$ exists for all $\varepsilon > 0$ does not exist for any $\varepsilon > 0$ exists for some $\varepsilon > 0$ none of the above is true

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 57 views

+4 votes

3 answers

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ISI2015-MMA-39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are $1,1,4$ $1,4,4$ $0,1,4$ $0,4,4$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 157 views

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ISI2015-MMA-40
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ by $$\textbf{A} = \begin{pmatrix} x_1^2+y_1^2 & x_1x_2 + y_1 y_2 & x_1x_3+y_1y_3 & x_1x_4+ ... ; x_4x_2+y_4y_2 & x_4x_3+y_4y_3 & x_4^2+y_4^2 \end{pmatrix}$$ Then rank$(\textbf{A})$ equals $1$ or $2$ $0$ $4$ $2$ or $3$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 62 views

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ISI2015-MMA-42
Let $\lambda_1, \lambda_2, \lambda_3$ denote the eigenvalues of the matrix $$A \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & - \sin t & \cos t \end{pmatrix}.$$ If $\lambda_1+\lambda_2+\lambda_3 = \sqrt{2}+1$, then the set of possible values of $t, \: - \pi ... {\pi}{4} \}$ $\{ - \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{ - \frac{\pi}{3}, \frac{\pi}{3} \}$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 95 views

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1 answer

25

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ISI2015-MMA-43
The values of $\eta$ for which the following system of equations $$\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$$ has a solution are $\eta = 1, -2$ $\eta = -1, -2$ $\eta = 3, -3$ $\eta = 1, 2$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 52 views

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1 answer

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ISI2015-MMA-44
Let $P_1$, $P_2$ and $P_3$ denote, respectively, the planes defined by $$\begin{array} {} a_1x +b_1y+c_1z=\alpha _1 \\ a_2x +b_2y+c_2z=\alpha _2 \\ a_3x +b_3y+c_3z=\alpha _3 \end{array}$$ It is given that $P_1$, $P_2$ and $P_3$ ... 4$ then the planes do not have any common point of intersection intersect at a unique point intersect along a straight line intersect along a plane

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 50 views

+1 vote

1 answer

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ISI2015-MMA-61
Let $$ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix} \text{ and } B=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}.$$ Then there exists a ... =BC=CB$ there is no matrix $C$ such that $A=BC$ there exists a matrix $C$ such that $A=BC$, but $A \neq CB$ there is no matrix $C$ such that $A=CB$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 72 views

+1 vote

1 answer

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ISI2015-MMA-62
If the matrix $A = \begin{bmatrix} a & 1 \\ 2 & 3 \end{bmatrix}$ has $1$ as an eigenvalue, then $\textit{trace}(A)$ is $4$ $5$ $6$ $7$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 98 views

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1 answer

29

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ISI2015-MMA-63
Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ is $\begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{ ... \\ - \sin^{30} \theta & \cos^{30} \theta \end{pmatrix}$ $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

asked Sep 23, 2019 in Linear Algebra by Arjun Veteran (436k points) 48 views

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2 answers

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ISI2015-DCG-3
The value of $\begin{vmatrix} 1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d \end{vmatrix}$ is $abcd(1+\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})$ $abcd(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d})$ $1+\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}$ None of these

asked Sep 18, 2019 in Linear Algebra by gatecse Boss (17.7k points) 81 views

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