Question 1 |
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 1 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 2 |
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 2 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 3 |
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 3 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 4 |
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 4 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 5 |
Multiplication of real valued square matrices of same dimension is
associative | |
commutative | |
always positive definite | |
not always possible to compute |
Question 5 Explanation:
Matrix multiplication is associative.
Question 6 |
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 6 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]
Question 7 |
In matrix equation [A]{X}={R},
[A]=\begin{bmatrix} 4 & 8 & 4\\ 8& 16 & -4\\ 4& -4 & 15 \end{bmatrix}, \{X\}=\begin{Bmatrix} 2\\ 1\\ 4 \end{Bmatrix} and \{R\}=\begin{Bmatrix} 32\\ 16\\ 64 \end{Bmatrix}.
One of the eigenvalues of matrix [A] is
[A]=\begin{bmatrix} 4 & 8 & 4\\ 8& 16 & -4\\ 4& -4 & 15 \end{bmatrix}, \{X\}=\begin{Bmatrix} 2\\ 1\\ 4 \end{Bmatrix} and \{R\}=\begin{Bmatrix} 32\\ 16\\ 64 \end{Bmatrix}.
One of the eigenvalues of matrix [A] is
4 | |
8 | |
15 | |
16 |
Question 7 Explanation:
Given that AX=R
\begin{array}{l} \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=\left[\begin{array}{l} 32 \\ 16 \\ 64 \end{array}\right] \\ \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=16\left[\begin{array}{c} 32 \\ 4 \end{array}\right](\because A X=\lambda X) \end{array}
\therefore One of eigen value of the given matrix A is given by \lambda=16
\begin{array}{l} \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=\left[\begin{array}{l} 32 \\ 16 \\ 64 \end{array}\right] \\ \Rightarrow\left[\begin{array}{ccc} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{array}\right]\left[\begin{array}{l} 2 \\ 1 \\ 4 \end{array}\right]=16\left[\begin{array}{c} 32 \\ 4 \end{array}\right](\because A X=\lambda X) \end{array}
\therefore One of eigen value of the given matrix A is given by \lambda=16
Question 8 |
The set of equations
x+y+z=1 \\ax-ay+3z=5 \\5x-3y+az=6
has infinite solutions, if a=
x+y+z=1 \\ax-ay+3z=5 \\5x-3y+az=6
has infinite solutions, if a=
-3 | |
3 | |
4 | |
-4 |
Question 8 Explanation:
The given system of equations can be expressed in the matrix form:
\left[\begin{array}{ccc} 1 & 1 & 1 \\ a & -a & 3 \\ 5 & -3 & a \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 1 \\ 5 \\ 6 \end{array}\right]
\mathrm{AX}=\mathrm{B} form
\text { Where } A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ a & -a & 3 \\ 5 & -3 & a \end{array}\right] X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \& B=\left[\begin{array}{c} 1 \\ 5 \\ 6 \end{array}\right]
The augmented matrix is
[A / B]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ a & -a & 3 & 5 \\ 5 & -3 & a & 6 \end{array}\right]
\mathrm{R}_{2}-\mathrm{aR}_{1} \& \mathrm{R}_{3}-5 \mathrm{R}_{1}
[\mathrm{A} / \mathrm{B}]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ 0 & -2 \mathrm{a} & 3-\mathrm{a} & 5-\mathrm{a} \\ 0 & -8 & \mathrm{a}-5 & 1 \end{array}\right]
\mathrm{aR}_{3}-4 \mathrm{R}_{2}
[\mathrm{A} / \mathrm{B}]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ 0 & -2 \mathrm{a} & 3-\mathrm{a} & 5-\mathrm{a} \\ 0 & 0 & \mathrm{a}^{2}-\mathrm{a}-12 & 5 \mathrm{a}-20 \end{array}\right]
For infinite solutions
\rho(\mathrm{A} / \mathrm{B})=\rho(\mathrm{A}) \lt Number of unknowns
\Rightarrow a^{2}-a-12=0 \& 5 a-20=0
\Rightarrow a=4
\left[\begin{array}{ccc} 1 & 1 & 1 \\ a & -a & 3 \\ 5 & -3 & a \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} 1 \\ 5 \\ 6 \end{array}\right]
\mathrm{AX}=\mathrm{B} form
\text { Where } A=\left[\begin{array}{ccc} 1 & 1 & 1 \\ a & -a & 3 \\ 5 & -3 & a \end{array}\right] X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \& B=\left[\begin{array}{c} 1 \\ 5 \\ 6 \end{array}\right]
The augmented matrix is
[A / B]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ a & -a & 3 & 5 \\ 5 & -3 & a & 6 \end{array}\right]
\mathrm{R}_{2}-\mathrm{aR}_{1} \& \mathrm{R}_{3}-5 \mathrm{R}_{1}
[\mathrm{A} / \mathrm{B}]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ 0 & -2 \mathrm{a} & 3-\mathrm{a} & 5-\mathrm{a} \\ 0 & -8 & \mathrm{a}-5 & 1 \end{array}\right]
\mathrm{aR}_{3}-4 \mathrm{R}_{2}
[\mathrm{A} / \mathrm{B}]=\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1 \\ 0 & -2 \mathrm{a} & 3-\mathrm{a} & 5-\mathrm{a} \\ 0 & 0 & \mathrm{a}^{2}-\mathrm{a}-12 & 5 \mathrm{a}-20 \end{array}\right]
For infinite solutions
\rho(\mathrm{A} / \mathrm{B})=\rho(\mathrm{A}) \lt Number of unknowns
\Rightarrow a^{2}-a-12=0 \& 5 a-20=0
\Rightarrow a=4
Question 9 |
Consider the matrix
P=\begin{bmatrix} 1 & 1 &0 \\ 0&1 &1 \\ 0& 0 & 1 \end{bmatrix}
The number of distinct eigenvalues of P is
P=\begin{bmatrix} 1 & 1 &0 \\ 0&1 &1 \\ 0& 0 & 1 \end{bmatrix}
The number of distinct eigenvalues of P is
0 | |
1 | |
2 | |
3 |
Question 9 Explanation:
\text { Given: } A=\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]
It is an upper triangular matrix. It's diagonal elements are eigen values.
The eigen values of the matrix are 1, 1, 1.
\therefore Number of distinct eigen values = 1
Hence, option (B) is correct.
It is an upper triangular matrix. It's diagonal elements are eigen values.
The eigen values of the matrix are 1, 1, 1.
\therefore Number of distinct eigen values = 1
Hence, option (B) is correct.
Question 10 |
The problem of maximizing z=x_{1}-x_{2} subject to constraints x_{1}+x_{2}\leq 10, x_{1}\geq 0,x_{2}\geq 0 and x_{2}\leq 5 has
no solution | |
one solution | |
two solutions | |
more than two solutions |
Question 10 Explanation:
\begin{aligned} \text{Max} ., Z=x_{1}-x_{2}\\ Constratins : x_{1}+x_{2} \leq 10 \\ x_{1} \geq 0 ; x_{2} \geq 0 & \text { and } x_{2} \leq 5 \\ Z(0,5)=0-5 &=-5 \\ Z(5,5)=5-5=0 & \\ Z(10,0)=10-0=& 10 \\ Z_{\max }=10 \text { at }(10,0) \end{aligned}
\therefore The problem has one solution
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 👍