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Recent questions and answers in Linear Algebra

1 vote
2 answers
1
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 $\begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix}$ ... $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
answered Jul 6 in Linear Algebra rishabhjain18 269 views
20 votes
4 answers
2
Let $u$ and $v$ be two vectors in $\mathbf{R}^{2}$ 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 $v$? $2$ $\frac{1}{2}$ $1$ $\frac{ -1}{2}$
answered Jun 19 in Linear Algebra HitechGa 8.5k views
7 votes
4 answers
3
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 __________
answered Jun 18 in Linear Algebra HitechGa 1.6k views
26 votes
4 answers
4
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_________________.
answered Jun 18 in Linear Algebra HitechGa 5.8k views
27 votes
3 answers
5
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 $\leq a +b$ $\leq \max(a, b)$ $\leq \min(M-a, N-b)$ $\leq \min(a, b)$
answered Jun 18 in Linear Algebra HitechGa 5.5k views
19 votes
6 answers
6
Consider the following statements: S1: The sum of two singular $n \times n$ matrices may be non-singular S2: The sum of two $n \times n$ non-singular matrices may be singular Which one of the following statements is correct? $S1$ and $S2$ both are true $S1$ is true, $S2$ is false $S1$ is false, $S2$ is true $S1$ and $S2$ both are false
answered Jun 18 in Linear Algebra HitechGa 4.7k views
31 votes
6 answers
7
The eigen vector $(s)$ of the matrix $\begin{bmatrix} 0 &0 &\alpha\\ 0 &0 &0\\ 0 &0 &0 \end{bmatrix},\alpha \neq 0$ is (are) $(0,0,\alpha)$ $(\alpha,0,0)$ $(0,0,1)$ $(0,\alpha,0)$
answered Jun 16 in Linear Algebra HitechGa 5.5k views
15 votes
7 answers
8
Let $P_{1},P_{2},\ldots,P_{n}$ be $n$ points in the $xy-$plane such that no three of them are collinear. For every pair of points $P_{i}$ and $P_{j}$, let $L_{ij}$ be the line passing through them. Let $L_{ab}$ be the line with the steepest ... the largest or the smallest $y$-coordinate among all the points The difference between $x$-coordinates $P_{a}$ and $P_{b}$ is minimum None of the above
answered Jun 16 in Linear Algebra HitechGa 2.4k views
25 votes
6 answers
9
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 $\left\{a\left(4,2,1\right) \mid a \neq 0, a \in \mathbb{R}\right\}$ ... $\left\{a\left(- \sqrt{2},0,1\right) \mid a \neq 0, a \in \mathbb{R}\right\}$
answered Jun 16 in Linear Algebra Divya7 8.2k views
20 votes
2 answers
10
If $A = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & -1 & 0 & -1 \\ 0 & 0 & i & i \\ 0 & 0 & 0 & -i \end{pmatrix}$ the matrix $A^4$, calculated by the use of Cayley-Hamilton theorem or otherwise, is _______
answered Jun 3 in Linear Algebra sauravgahlawat 3.3k views
6 votes
5 answers
11
Let $A$ be an $n\times n$ invertible matrix with real entries whose row sums are all equal to $c$. Consider the following statements: Every row in the matrix $2A$ sums to $2c$. Every row in the matrix $A^{2}$ sums to $c^{2}$. Every row in the matrix $A^{-1}$ sums ... statement $(1)$ and $(2)$ are correct but not necessarily statement $(3)$ all the three statements $(1), (2),$ and $(3)$ are correct
answered May 29 in Linear Algebra Yawar Rmir 1.6k views
1 vote
2 answers
12
2 votes
2 answers
13
Let $A=\begin{pmatrix} 1 & 1 & 0\\ 0 & a & b\\1 & 0 & 1 \end{pmatrix}$. Then $A^{-1}$ does not exist if $(a,b)$ is equal to $(1,-1)$ $(1,0)$ $(-1,-1)$ $(0,1)$
answered May 6 in Linear Algebra eshita1997 197 views
1 vote
2 answers
14
let $\left | A \right|=8$ ,$\left | B \right|=3$ ,$\left | C \right|=6$ then what will be value of AB$^{T}$C$^{-1}$ A) 144 B) 0 C) 4 D) 14
answered Apr 2 in Linear Algebra heisenberggg 501 views
1 vote
1 answer
15
The value of $p$ such that the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is an eigen vector of the matrix $\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{bmatrix}$ is $15$ $16$ $17$ $18$
answered Mar 28 in Linear Algebra Asim Siddiqui 4 802 views
56 votes
4 answers
16
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? $-5$ $-7$ $2$ $1$
answered Mar 24 in Linear Algebra Bikki_gupta 8.8k views
20 votes
8 answers
17
What are the eigenvalues of the matrix $P$ given below $P= \begin{pmatrix} a &1 &0 \\ 1& a& 1\\ 0&1 &a \end{pmatrix}$ $a, a -√2, a + √2$ $a, a, a$ $0, a, 2a$ $-a, 2a, 2a$
answered Mar 24 in Linear Algebra Bikki_gupta 3.9k views
23 votes
5 answers
18
Obtain the eigen values of the matrix$A=\begin {bmatrix} 1 & 2 & 34 & 49 \\ 0 & 2 & 43 & 94 \\ 0 & 0 & -2 & 104 \\ 0 & 0 & 0 & -1 \end{bmatrix}$
answered Mar 24 in Linear Algebra Bikki_gupta 2.3k views
19 votes
6 answers
19
The determinant of the matrix given below is $\begin{bmatrix} 0 &1 &0 &2 \\ -1& 1& 1& 3\\ 0&0 &0 & 1\\ 1& -2& 0& 1 \end{bmatrix}$ $-1$ $0$ $1$ $2$
answered Mar 24 in Linear Algebra Bikki_gupta 5.3k views
2 votes
3 answers
20
Let $A$ = $\begin{bmatrix} 1 & 0 & -1 \\-1 & 2 & 0 \\ 0 & 0 & -2 \end{bmatrix}$ and $B$ = $A^{3} - A^{2} -4A +5I$ where $I$ is the $3\times 3$ identity matrix. The determinant of $B$ is ________ (up to $1$ decimal place).
answered Mar 23 in Linear Algebra Bikki_gupta 1.8k views
15 votes
5 answers
21
The determinant of the matrix $\begin{bmatrix}2 &0 &0 &0 \\ 8& 1& 7& 2\\ 2& 0&2 &0 \\ 9&0 & 6 & 1 \end{bmatrix}$ $4$ $0$ $15$ $20$
answered Mar 20 in Linear Algebra Bikki_gupta 3.3k views
13 votes
3 answers
22
The matrices $\begin{bmatrix} \cos\theta && -\sin\theta \\ \sin\theta && \cos\theta \end{bmatrix}$ and $\begin{bmatrix} a && 0\\ 0&& b \end{bmatrix}$ commute under multiplication if $a=b \text{ or } \theta = n\pi, n$ an integer always never if $a \cos\theta = b \sin\theta$
answered Mar 19 in Linear Algebra Bikki_gupta 2.6k views
2 votes
2 answers
23
2 votes
2 answers
24
If $\begin{vmatrix} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{vmatrix} = k(10!)(11!)(12!)$, then the value of $k$ is $1$ $2$ $3$ $4$
answered Feb 23 in Linear Algebra Auditi 158 views
23 votes
3 answers
25
How many of the following matrices have an eigenvalue 1? $\left[\begin{array}{cc}1 & 0 \\0 & 0 \end{array} \right]\left[\begin{array}{cc}0 & 1 \\0 & 0 \end{array} \right] \left[\begin{array}{cc}1 & -1 \\1 & 1 \end{array} \right]$ and $\left[\begin{array}{cc}-1 & 0 \\1 & -1 \end{array} \right]$ one two three four
answered Jan 16 in Linear Algebra Surya_Dev Chaturvedi 4.4k views
1 vote
1 answer
26
consider a system of linear equation where AMxN XNx1 =BMx1 TRUE OR FALSE Q1 IF B=0 AND DETERMINENT OF A i.e |A| IS NOT EQUAL TO ZERO THEN IT MEANS UNIQUE SOLUTION?? Q2 IF B NOT EQUAL TO 0 AND DETERMINENT OF A i.e |A| IS NOT EQUAL TO ZERO THEN IT MEANS INFINITE MANY SOLUTION SOLUTION?? Q3 IF B IS NOT EQUAL TO 0 AND M<N THEN IT MEANS NO UNIQUE SOLUTION??
answered Dec 31, 2020 in Linear Algebra reboot 153 views
1 vote
2 answers
27
Let AX=B be a system of n linear equations in n unknown with integer coefficient and the components of B are all integer. Consider the following (1)det(A)=1 (2)det(A)=0 (3)Solution X has integer entries (4)Solution X does not have all integer entries For the given system of linear ... 1, then 3 holds true (c)If 1, then 4 holds true (d)If 2, then 3 holds true I think (d) should be the answer.
answered Dec 31, 2020 in Linear Algebra reboot 700 views
0 votes
3 answers
28
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 $P_3$ intersect ... 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
answered Dec 31, 2020 in Linear Algebra reboot 329 views
1 vote
2 answers
29
answered Dec 26, 2020 in Linear Algebra eshita1997 170 views
29 votes
4 answers
30
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 S3: $MX = 0$ has a nontrivial solution S4: $M$ has an inverse $S3$ and $S2$ $S1$ and $S4$ $S1$ and $S3$ $S1, S2$ and $S3$
answered Dec 14, 2020 in Linear Algebra Ashley Varghese Joy 4.8k views
0 votes
2 answers
31
The matrices $\begin{bmatrix} \cos\theta &-\sin \theta \\ \sin \theta & cos \theta \end{bmatrix}$ and $\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$ commute under the multiplication if $a=b \text{(or)} \theta =n\pi, \: n$ is an integer always never if $a\cos \theta \neq b\sin \theta$
asked Apr 2, 2020 in Linear Algebra Lakshman Patel RJIT 205 views
0 votes
1 answer
32
0 votes
1 answer
33
Consider three vectors $x=\begin{bmatrix}1\\2 \end{bmatrix}, y=\begin{bmatrix}4\\8 \end{bmatrix},z=\begin{bmatrix}3\\1 \end{bmatrix}$. Which of the folowing statements is true $x$ and $y$ are linearly independent $x$ and $y$ are linearly dependent $x$ and $z$ are linearly dependent $y$ and $z$ are linearly dependent
asked Apr 2, 2020 in Linear Algebra Lakshman Patel RJIT 151 views
0 votes
0 answers
34
If product of matrix $A=\begin{bmatrix}\cos^{2}\theta &\cos \theta \sin \theta \\ \cos \theta \sin \theta &\sin ^{2} \theta& \end{bmatrix}$ and $B=\begin{bmatrix}\cos^{2}\phi &\cos \phi \sin \phi \\ \cos \phi \sin \phi &\sin ^{2} \phi& \end{bmatrix}$ is a ... and $\phi$ differ by an odd multiple of $\pi$ even multiple of $\pi$ odd multiple of $\dfrac{\pi}{2}$ even multiple of $\dfrac{\pi}{2}$
asked Apr 2, 2020 in Linear Algebra Lakshman Patel RJIT 170 views
0 votes
1 answer
35
$M$ is a square matrix of order $’n’$ and its determinant value is $5.$ If all the elements of $M$ are multiplied by $2,$ its determinant value becomes $40.$ The value of $’n’$ is $2$ $3$ $5$ $4$
asked Apr 1, 2020 in Linear Algebra Lakshman Patel RJIT 204 views
0 votes
2 answers
36
0 votes
1 answer
37
0 votes
1 answer
38
0 votes
2 answers
39
If $A$ and $B$ are square matrices of size $n\times n$, then which of the following statements is not true? $\det(AB)=\det(A) \det(B)$ $\det(kA)=k^n \det(A)$ $\det(A+B)=\det(A)+\det(B)$ $\det(A^T)=1/\det(A^{-1})$
asked Mar 31, 2020 in Linear Algebra Lakshman Patel RJIT 2.1k views
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