Question 1 |
If the sum and product of eigenvalues of a 2 \times 2
real matrix \begin{bmatrix}
3 & p\\
p & q
\end{bmatrix} are 4 and -1 respectively, then |p|
is ______ (in integer).
4 | |
2 | |
6 | |
8 |
Question 1 Explanation:
From the property of eigen values,
Sum of eigen values = Trace of matrix
4=3+q
q=1
Product of eigen values = Determinant
\begin{aligned} -1&=\begin{vmatrix} 3 &p \\ p& q \end{vmatrix}\\ 3q-p^2&=-1\\ 3-p^2&=-1\\ p^2&=4\\ p&=\pm 2\\ \Rightarrow |p|&=2 \end{aligned}
Sum of eigen values = Trace of matrix
4=3+q
q=1
Product of eigen values = Determinant
\begin{aligned} -1&=\begin{vmatrix} 3 &p \\ p& q \end{vmatrix}\\ 3q-p^2&=-1\\ 3-p^2&=-1\\ p^2&=4\\ p&=\pm 2\\ \Rightarrow |p|&=2 \end{aligned}
Question 2 |
A is a 3\times 5 real matrix of rank 2. For the set of
homogeneous equations Ax = 0, where 0 is a zero
vector and x is a vector of unknown variables,
which of the following is/are true?
The given set of equations will have a unique solution. | |
The given set of equations will be satisfied by a zero vector of appropriate size. | |
The given set of equations will have infinitely many solutions. | |
The given set of equations will have many but a finite number of solutions. |
Question 2 Explanation:
Zero solution is always a solution of Ax = 0.
Option (b) is correct.
Given A is 3x5 real matrix and r(A) = 2 and Ax = 0 is a system of homogeneous linear equations since r(A) \lt number of unknown the system has infinite solution option (c) is also correct.
Option (b) is correct.
Given A is 3x5 real matrix and r(A) = 2 and Ax = 0 is a system of homogeneous linear equations since r(A) \lt number of unknown the system has infinite solution option (c) is also correct.
Question 3 |
The system of linear equations in real (x, y) given
by
(x\;\;y)\begin{bmatrix} 2 & 5-2\alpha \\ \alpha & 1 \end{bmatrix} =(0\;\;0)
involves a real parameter \alpha and has infinitely many non-trivial solutions for special value(s) of \alpha . Which one or more among the following options is/ are non-trivial solution(s) of (x,y) for such special value(s) of \alpha ?
(x\;\;y)\begin{bmatrix} 2 & 5-2\alpha \\ \alpha & 1 \end{bmatrix} =(0\;\;0)
involves a real parameter \alpha and has infinitely many non-trivial solutions for special value(s) of \alpha . Which one or more among the following options is/ are non-trivial solution(s) of (x,y) for such special value(s) of \alpha ?
x=2,y=-2 | |
x=-1,y=4 | |
x=1,y=1 | |
x=4,y=-2 |
Question 3 Explanation:
\begin{aligned}
\begin{pmatrix}
x &y
\end{pmatrix}\begin{pmatrix}
2 &5-2\alpha \\
\alpha & 1
\end{pmatrix}&=\begin{pmatrix}
0 &0
\end{pmatrix}\\
2x+\alpha y&=0\\
(5-2\alpha )x+y&=0\\
\therefore \begin{pmatrix}
2 & \alpha \\
5-2\alpha & 1
\end{pmatrix}\begin{pmatrix}
x\\y
\end{pmatrix}&=\begin{pmatrix}
0\\0
\end{pmatrix} \;\;\; . . . . (1)
\end{aligned}
To get infinite number of non-trivial solutions \begin{aligned} \begin{vmatrix} 2 &\alpha \\ 5-2\alpha & 1 \end{vmatrix}&=0\\ 2-(5\alpha -2\alpha ^2)&=0\\ (2\alpha -1)(\alpha -2)&=0\\ \therefore \alpha =\frac{1}{2},\alpha &=2\\ \end{aligned}
At \alpha =\frac{1}{2} ; eq. (1) gives (4x+y)=0 . . .(2)
option (B) is satisfied by (2)
At \alpha =2 ; eq. (1) gives (x+y)=0 . . .(3)
option (A) is satisfied by (3)
Both options (A) and (B) are correct
To get infinite number of non-trivial solutions \begin{aligned} \begin{vmatrix} 2 &\alpha \\ 5-2\alpha & 1 \end{vmatrix}&=0\\ 2-(5\alpha -2\alpha ^2)&=0\\ (2\alpha -1)(\alpha -2)&=0\\ \therefore \alpha =\frac{1}{2},\alpha &=2\\ \end{aligned}
At \alpha =\frac{1}{2} ; eq. (1) gives (4x+y)=0 . . .(2)
option (B) is satisfied by (2)
At \alpha =2 ; eq. (1) gives (x+y)=0 . . .(3)
option (A) is satisfied by (3)
Both options (A) and (B) are correct
Question 4 |
If A=\begin{bmatrix}
10 &2k+5 \\
3k-3 & k+5
\end{bmatrix} is a symmetric matrix, the
value of k is ___________.
8 | |
5 | |
-0.4 | |
\frac{1+\sqrt{1561}}{12} |
Question 4 Explanation:
A=\begin{bmatrix}
10 & 2k+5\\
3k-3 & k+5
\end{bmatrix}
(2k + 5) = (3k - 3)
k=8
(2k + 5) = (3k - 3)
k=8
Question 5 |
Consider an n x n matrix A and a non-zero n x 1 vector p. Their product Ap=\alpha ^2p, where \alpha \in \mathbb{R} and \alpha \notin \{-1,0,1\}. Based on the given information, the eigen value of A^2 is:
\alpha | |
\alpha ^2 | |
\sqrt{\alpha } | |
\alpha ^4 |
Question 5 Explanation:
Given, A P=\alpha^{2} P
By comparison with A X=\lambda X \Rightarrow
\Rightarrow \quad \lambda=\alpha^{2}
Hence, eigen value of A is \alpha^{2}, so eigen value of A^{2} is \alpha^{4}.
By comparison with A X=\lambda X \Rightarrow
\Rightarrow \quad \lambda=\alpha^{2}
Hence, eigen value of A is \alpha^{2}, so eigen value of A^{2} is \alpha^{4}.
Question 6 |
Consider a vector p in 2-dimensional space. Let its direction (counter- clockwise angle with the positive x-axis) be \theta. Let p be an eigenvector of a 2 \times 2 matrix A with corresponding eigenvalue \lambda, \; \lambda, > 0. If we denote the magnitude of a vector v by ||v||, identify the VALID statement regarding p', where p'=Ap.
Direction of p'=\lambda \theta ,\; ||p'||=||p|| | |
Direction of p'= \theta ,\; ||p'||=\lambda||p|| | |
Direction of p'=\lambda \theta ,\; ||p'||=\lambda||p|| | |
Direction of p'=\theta ,\; ||p'||=||p|| / \lambda |
Question 6 Explanation:
\because A is a 2 \times 2 matrix and P is the eigen vector of matrix A with corresponding eigen value \lambda
\begin{aligned} \text{Given}:\quad P^{\prime} &=P \\ A P &=\lambda P \\ P^{\prime} &=\lambda P\\ \text{Hence},\quad \left\|P^{\prime}\right\|&=\|\lambda P\|=\lambda|P|\\ \end{aligned}
But direction of vector P^{\prime} will be same as vector P.
\begin{aligned} \text{Given}:\quad P^{\prime} &=P \\ A P &=\lambda P \\ P^{\prime} &=\lambda P\\ \text{Hence},\quad \left\|P^{\prime}\right\|&=\|\lambda P\|=\lambda|P|\\ \end{aligned}
But direction of vector P^{\prime} will be same as vector P.
Question 7 |
Let I be a 100 dimensional identity matrix and E be the set of its distinct (no value
appears more than once in E ) real eigenvalues. The number of elements in E is ______.
1 | |
100 | |
10 | |
0 |
Question 7 Explanation:
I_{100}
Eigen values of I \rightarrow \underset{100 \text { times }}{\underbrace{1,1, \ldots \ldots \ldots ., 1}}
Set of distributed eigen value E=\{1\}
Number of elements in E=1
Eigen values of I \rightarrow \underset{100 \text { times }}{\underbrace{1,1, \ldots \ldots \ldots ., 1}}
Set of distributed eigen value E=\{1\}
Number of elements in E=1
Question 8 |
A matrix P is decomposed into its symmetric part S and skew symmetric part V. If
S=\begin{pmatrix} -4 &4 &2 \\ 4& 3 & 7/2\\ 2& 7/2 & 2 \end{pmatrix}, V=\begin{pmatrix} 0 &-2 &3 \\ 2& 0 & 7/2\\ -3& -7/2 & 0 \end{pmatrix},
Then matrix P is
S=\begin{pmatrix} -4 &4 &2 \\ 4& 3 & 7/2\\ 2& 7/2 & 2 \end{pmatrix}, V=\begin{pmatrix} 0 &-2 &3 \\ 2& 0 & 7/2\\ -3& -7/2 & 0 \end{pmatrix},
Then matrix P is
A | |
B | |
C | |
D |
Question 8 Explanation:
\begin{array}{l} S=\frac{P+P^{T}}{2} \\ V=\frac{P-P^{T}}{2} \\ P=S+V=\left(\begin{array}{ccc} -4 & 2 & 5 \\ 6 & 3 & 7 \\ -1 & 0 & 2 \end{array}\right) \end{array}
Question 9 |
Multiplication of real valued square matrices of same dimension is
associative | |
commutative | |
always positive definite | |
not always possible to compute |
Question 9 Explanation:
Matrix multiplication is associative.
Question 10 |
The transformation matrix for mirroring a point in x-y plane about the line y=x is given by
\begin{bmatrix} 1 & 0\\ 0&-1 \end{bmatrix} | |
\begin{bmatrix} -1 & 0\\ 0&1 \end{bmatrix} | |
\begin{bmatrix} 0 & 1\\ 1&0 \end{bmatrix} | |
\begin{bmatrix} 0 & -1\\ -1&0 \end{bmatrix} |
Question 10 Explanation:
The transformation matrix for mirroring a point in x-y plane about the line y = x is
\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]
\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]
There are 10 questions to complete.
56 no please check
Dear Sangram Kumar
Thank you for your suggestions. We have updated the correction suggested by You.
Great job sir 👍
In question a12 should be -i
In question 24
The answer for question no 4 should be A and it is marked as B
Option B is correct answer.
Sir in question no 24 first row second couloum should be -i
In the Question no 1 check matrix once again
Question 1 is wrong the matrix in the question is wrong