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Webpage for Linear Algebra
Recent questions tagged linear-algebra
12
votes
2
answers
541
GATE CSE 1995 | Question: 2.13
A unit vector perpendicular to both the vectors $a=2i-3j+k$ and $b=i+j-2k$ is: $\frac{1}{\sqrt{3}} (i+j+k)$ $\frac{1}{3} (i+j-k)$ $\frac{1}{3} (i-j-k)$ $\frac{1}{\sqrt{3}} (i+j-k)$
Kathleen
asked
in
Linear Algebra
Oct 8, 2014
by
Kathleen
2.8k
views
gate1995
linear-algebra
normal
vector-space
21
votes
6
answers
542
GATE CSE 1995 | Question: 1.24
The rank of the following $(n+1) \times (n+1)$ matrix, where $a$ ... $1$ $2$ $n$ Depends on the value of $a$
Kathleen
asked
in
Linear Algebra
Oct 8, 2014
by
Kathleen
3.8k
views
gate1995
linear-algebra
matrix
normal
rank-of-matrix
16
votes
3
answers
543
GATE CSE 1994 | Question: 3.12
Find the inverse of the matrix $\begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix}$
Kathleen
asked
in
Linear Algebra
Oct 6, 2014
by
Kathleen
3.5k
views
gate1994
linear-algebra
matrix
easy
descriptive
16
votes
2
answers
544
GATE CSE 1994 | Question: 1.9
The rank of matrix $\begin{bmatrix} 0 & 0 & -3 \\ 9 & 3 & 5 \\ 3 & 1 & 1 \end{bmatrix}$ is: $0$ $1$ $2$ $3$
Kathleen
asked
in
Linear Algebra
Oct 4, 2014
by
Kathleen
4.1k
views
gate1994
linear-algebra
matrix
rank-of-matrix
easy
25
votes
2
answers
545
GATE CSE 1994 | Question: 1.2
Let $A$ and $B$ be real symmetric matrices of size $n \times n$. Then which one of the following is true? $AA'=I$ $A=A^{-1}$ $AB=BA$ $(AB)'=BA$
Kathleen
asked
in
Linear Algebra
Oct 4, 2014
by
Kathleen
6.1k
views
gate1994
linear-algebra
normal
matrix
22
votes
4
answers
546
GATE CSE 1997 | Question: 4.2
Let $A=(a_{ij})$ be an $n$-rowed square matrix and $I_{12}$ be the matrix obtained by interchanging the first and second rows of the $n$-rowed Identity matrix. Then $AI_{12}$ is such that its first Row is the same as its second row Row is the same as the second row of $A$ Column is the same as the second column of $A$ Row is all zero
Kathleen
asked
in
Linear Algebra
Sep 29, 2014
by
Kathleen
3.5k
views
gate1997
linear-algebra
easy
matrix
16
votes
1
answer
547
GATE CSE 1997 | Question: 1.3
The determinant of the matrix $\begin{bmatrix} 6 & -8 & 1 & 1 \\ 0 & 2 & 4 & 6 \\ 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & -1 \end{bmatrix}$ $11$ $-48$ $0$ $-24$
Kathleen
asked
in
Linear Algebra
Sep 29, 2014
by
Kathleen
2.6k
views
gate1997
linear-algebra
normal
determinant
22
votes
4
answers
548
GATE CSE 2011 | Question: 40
Consider the matrix as given below. $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 0 & 0 & 3\end{bmatrix}$ Which one of the following options provides the CORRECT values of the eigenvalues of the matrix? $1, 4, 3$ $3, 7, 3$ $7, 3, 2$ $1, 2, 3$
go_editor
asked
in
Linear Algebra
Sep 29, 2014
by
go_editor
3.7k
views
gatecse-2011
linear-algebra
eigen-value
easy
28
votes
2
answers
549
GATE CSE 2014 Set 3 | Question: 5
If $V_1$ and $V_2$ are $4$-dimensional subspaces of a $6$-dimensional vector space $V$, then the smallest possible dimension of $V_1 \cap V_2$ is _____.
go_editor
asked
in
Linear Algebra
Sep 28, 2014
by
go_editor
8.1k
views
gatecse-2014-set3
linear-algebra
vector-space
normal
numerical-answers
34
votes
3
answers
550
GATE CSE 2014 Set 3 | Question: 4
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 ... eigenvalues are positive. If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
go_editor
asked
in
Linear Algebra
Sep 28, 2014
by
go_editor
8.8k
views
gatecse-2014-set3
linear-algebra
eigen-value
normal
77
votes
9
answers
551
GATE CSE 2014 Set 2 | Question: 47
The product of the non-zero eigenvalues of the matrix is ____ $\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{pmatrix}$
go_editor
asked
in
Linear Algebra
Sep 28, 2014
by
go_editor
28.5k
views
gatecse-2014-set2
linear-algebra
eigen-value
normal
numerical-answers
32
votes
6
answers
552
GATE CSE 2014 Set 2 | Question: 4
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 ______.
go_editor
asked
in
Linear Algebra
Sep 28, 2014
by
go_editor
9.5k
views
gatecse-2014-set2
linear-algebra
numerical-answers
easy
determinant
46
votes
4
answers
553
GATE CSE 2014 Set 1 | Question: 5
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 ___________
go_editor
asked
in
Linear Algebra
Sep 26, 2014
by
go_editor
10.8k
views
gatecse-2014-set1
linear-algebra
eigen-value
numerical-answers
normal
34
votes
5
answers
554
GATE CSE 2014 Set 1 | Question: 4
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 ______________
go_editor
asked
in
Linear Algebra
Sep 26, 2014
by
go_editor
9.2k
views
gatecse-2014-set1
linear-algebra
system-of-equations
numerical-answers
normal
4
votes
1
answer
555
GATE CSE 1998 | Question: 9
Derive the expressions for the number of operations required to solve a system of linear equations in $n$ unknowns using the Gaussian Elimination Method. Assume that one operation refers to a multiplication followed by an addition.
Kathleen
asked
in
Linear Algebra
Sep 26, 2014
by
Kathleen
1.9k
views
gate1998
linear-algebra
system-of-equations
descriptive
22
votes
4
answers
556
GATE CSE 1998 | Question: 2.2
Consider the following determinant $\Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}$ Which of the following is a factor of $\Delta$? $a+b$ $a-b$ $a+b+c$ $abc$
Kathleen
asked
in
Linear Algebra
Sep 26, 2014
by
Kathleen
5.2k
views
gate1998
linear-algebra
matrix
normal
18
votes
3
answers
557
GATE CSE 1998 | Question: 2.1
The rank of the matrix given below is: $\begin{bmatrix} 1 &4 &8 &7\\ 0 &0& 3 &0\\ 4 &2& 3 &1\\ 3 &12 &24 &21 \end{bmatrix}$ $3$ $1$ $2$ $4$
Kathleen
asked
in
Linear Algebra
Sep 26, 2014
by
Kathleen
5.0k
views
gate1998
linear-algebra
matrix
normal
20
votes
2
answers
558
GATE CSE 1998 | Question: 1.2
Consider the following set of equations $x+2y=5$ $4x+8y=12$ $3x+6y+3z=15$ This set has unique solution has no solution has finite number of solutions has infinite number of solutions
Kathleen
asked
in
Linear Algebra
Sep 26, 2014
by
Kathleen
5.3k
views
gate1998
linear-algebra
system-of-equations
easy
31
votes
3
answers
559
GATE CSE 2013 | Question: 3
Which one of the following does NOT equal $\begin{vmatrix} 1 & x & x^{2}\\ 1& y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} \quad ?$ $\begin{vmatrix} 1& x(x+1)& x+1\\ 1& y(y+1) & y+1\\ 1& z(z+1) & z+1 \end{vmatrix}$ ...
Arjun
asked
in
Linear Algebra
Sep 23, 2014
by
Arjun
6.4k
views
gatecse-2013
linear-algebra
normal
determinant
28
votes
4
answers
560
GATE CSE 2007 | Question: 27
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
Kathleen
asked
in
Linear Algebra
Sep 22, 2014
by
Kathleen
12.1k
views
gatecse-2007
linear-algebra
normal
vector-space
15
votes
3
answers
561
GATE CSE 2005 | Question: 49
What are the eigenvalues of the following $2\times 2$ matrix? $\left( \begin{array}{cc} 2 & -1\\ -4 & 5\end{array}\right)$ $-1$ and $1$ $1$ and $6$ $2$ and $5$ $4$ and $-1$
gatecse
asked
in
Linear Algebra
Sep 21, 2014
by
gatecse
4.5k
views
gatecse-2005
linear-algebra
eigen-value
easy
18
votes
2
answers
562
GATE CSE 2005 | Question: 48
Consider the following system of linear equations : $2x_1 - x_2 + 3x_3 = 1$ $3x_1 + 2x_2 + 5x_3 = 2$ $-x_1+4x_2+x_3 = 3$ The system of equations has no solution a unique solution more than one but a finite number of solutions an infinite number of solutions
gatecse
asked
in
Linear Algebra
Sep 21, 2014
by
gatecse
4.5k
views
gatecse-2005
linear-algebra
system-of-equations
normal
19
votes
2
answers
563
GATE CSE 2010 | Question: 29
Consider the following matrix $A = \left[\begin{array}{cc}2 & 3\\x & y \end{array}\right]$ If the eigenvalues of A are $4$ and $8$, then $x = 4$, $y = 10$ $x = 5$, $y = 8$ $x = 3$, $y = 9$ $x = -4$, $y =10$
gatecse
asked
in
Linear Algebra
Sep 21, 2014
by
gatecse
6.7k
views
gatecse-2010
linear-algebra
eigen-value
easy
32
votes
4
answers
564
GATE CSE 2004 | Question: 76
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 $\leq a +b$ $\leq \max(a, b)$ $\leq \min(M-a, N-b)$ $\leq \min(a, b)$
Kathleen
asked
in
Linear Algebra
Sep 19, 2014
by
Kathleen
7.1k
views
gatecse-2004
linear-algebra
normal
matrix
25
votes
3
answers
565
GATE CSE 2004 | Question: 71
How many solutions does the following system of linear equations have? $-x + 5y = -1$ $x - y = 2$ $x + 3y = 3$ infinitely many two distinct solutions unique none
Kathleen
asked
in
Linear Algebra
Sep 19, 2014
by
Kathleen
5.6k
views
gatecse-2004
linear-algebra
system-of-equations
normal
31
votes
4
answers
566
GATE CSE 2004 | Question: 27
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
Kathleen
asked
in
Linear Algebra
Sep 19, 2014
by
Kathleen
7.1k
views
gatecse-2004
linear-algebra
normal
matrix
24
votes
6
answers
567
GATE CSE 2004 | Question: 26
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 $2^X$) $\text{power} \left(2, n\right)$ $\text{power} \left(2, n^2\right)$ $\text{power} \left(2,\frac{ \left(n^2+ n \right) }{2}\right)$ $\text{power} \left(2, \frac{\left(n^2 - n\right)}{2}\right)$
Kathleen
asked
in
Linear Algebra
Sep 19, 2014
by
Kathleen
9.6k
views
gatecse-2004
linear-algebra
normal
matrix
38
votes
5
answers
568
GATE CSE 2006 | Question: 23
$F$ 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
Rucha Shelke
asked
in
Linear Algebra
Sep 17, 2014
by
Rucha Shelke
7.2k
views
gatecse-2006
linear-algebra
normal
matrix
28
votes
3
answers
569
GATE CSE 2003 | Question: 41
Consider the following system of linear equations ... linearly dependent. For how many values of $\alpha$, does this system of equations have infinitely many solutions? \(0\) \(1\) \(2\) \(3\)
Kathleen
asked
in
Linear Algebra
Sep 17, 2014
by
Kathleen
9.1k
views
gatecse-2003
linear-algebra
system-of-equations
normal
25
votes
5
answers
570
GATE CSE 2002 | Question: 5a
Obtain the eigen values of the matrix$A=\begin {bmatrix} 1 & 2 & 34 & 49 \\ 0 & 2 & 43 & 94 \\ 0 & 0 & -2 & 104 \\ 0 & 0 & 0 & -1 \end{bmatrix}$
Kathleen
asked
in
Linear Algebra
Sep 16, 2014
by
Kathleen
3.1k
views
gatecse-2002
linear-algebra
eigen-value
normal
descriptive
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