Linear Algebra

Question 1
Let \alpha ,\beta be two non-zero real numbers and v_1,v_2 be two non-zero real vectors of size 3 x 1. Suppose that v_1 and v_2 satisfy v_1^Tv_2=0,v_1^Tv_1=1, and v_2^Tv_2=1. Let A be the 3x3 matrix given by:
A=\alpha v_1v_1^T+ \beta v_2v_2^T
The eigenvalues of A are __________.
A
0,\alpha ,\beta
B
0,\alpha+\beta ,\alpha-\beta
C
0,\frac{\alpha +\beta}{2},\sqrt{\alpha \beta }
D
0,0,\sqrt{\alpha ^2+ \beta ^2}
GATE EC 2022   Engineering Mathematics
Question 1 Explanation: 
After multiply by v_1 on both sides,
\begin{aligned} Av_1&=\alpha v_1v_1^Tv_1+ \beta v_2v_2^Tv_1 Av_1&=\alpha v_1 \end{aligned}
\alpha is an eigen value of A.

After multiply by v_2 on both sides,
\begin{aligned} Av_2&=\alpha v_1v_1^Tv_2+ \beta v_2v_2^Tv_2 Av_2&=\beta v_2 \end{aligned}
\beta is an eigen value of A.
v_1v_1^T and v_2v_2^T both are singular matrieces.
Therefore, A is also a singular matrix.
|A|=0\Rightarrow \lambda _A=0
Hence \lambda _A= 0,\alpha ,\beta
Question 2
Consider a system of linear equations Ax=b, where
A=\begin{bmatrix} 1 & -\sqrt{2} & 3\\ -1& \sqrt{2}& -3 \end{bmatrix},b=\begin{bmatrix} 1\\ 3 \end{bmatrix}
This system of equations admits ______.
A
a unique solution for x
B
infinitely many solutions for x
C
no solutions for x
D
exactly two solutions for x
GATE EC 2022   Engineering Mathematics
Question 2 Explanation: 
Here equation will be
x-\sqrt{2}y+3z=1
-x+\sqrt{2}y-3z=3
therefore inconsistant solution i.e. there will not be any solution.
Question 3
A real 2\times2 non-singular matrix A with repeated eigenvalue is given as
A=\begin{bmatrix} x & -3.0\\ 3.0 & 4.0 \end{bmatrix}
where x is a real positive number. The value of x (rounded off to one decimal place) is ________________
A
5.2
B
18.7
C
10
D
6.8
GATE EC 2021   Engineering Mathematics
Question 3 Explanation: 
A=\left[\begin{array}{cc} x & -3 \\ 3 & 4 \end{array} \right]
Characteristic equation
\begin{aligned} |A-\lambda I| &=\left|\begin{array}{cc} x-\lambda & -3 \\ 3 & 4-\lambda \end{array}\right|=0 \\ \Rightarrow \quad \lambda^{2}-(4+x) \lambda+(4 x+9)&=0 \end{aligned}
Roots are repeating
\begin{aligned} \Rightarrow \qquad b^{2}-4 a c&=0\\ \Rightarrow \qquad \left(4+x^{2}\right)-4(4 x+9) &=0 \\ 16+x^{2}+8 x-16 x-20 &=0 \\ x &=10 \\ \Rightarrow \qquad x^{2}-8 x+20 &=0\\ x&=\frac{8 \pm \sqrt{64+80}}{2} \\ &=\frac{8 \pm 12}{2}=-2,10 \end{aligned}
Since, x is positive
\therefore x=10
Question 4
If the vectors (1.0,\:-1.0,\:2.0), (7.0,\:3.0,\:x) and (2.0,\:3.0,\:1.0) in \mathbb{R}^{3} are linearly dependent, the value of x is __________
A
8
B
9
C
4
D
2
GATE EC 2021   Engineering Mathematics
Question 4 Explanation: 
(1,-1,2)
(7,3, x) are linearly dependent when x=?
(2,3,1)
\begin{aligned} \Rightarrow\left|\begin{array}{ccc} 1 & -1 & 2 \\ 7 & 3 & x \\ 2 & 3 & 1 \end{array}\right| &=0 \Rightarrow 1(3-3 x)+1(7-2 x)+2(15)=0 \\ \Rightarrow \quad-5 x&=-40 \\ x&=8 \end{aligned}
Question 5
Consider the following system of linear equation.

x_1+2x_2=b_1;
2x_1+4x_2=b_2;
3x_1+7x_2=b_3;
3x_1+9x_2=b_4;

Which one of the following conditions ensures that a solution exists for the above system?
A
b_2=2b_1 and 6b_1-3b_3+b_4=0
B
b_3=2b_1 and 6b_1-3b_3+b_4=0
C
b_2=2b_1 and 3b_1-6b_3+b_4=0
D
b_3=2b_1 and 3b_1-6b_3+b_4=0
GATE EC 2020   Engineering Mathematics
Question 5 Explanation: 
Given:
x_{1}+2x_{2}=b_{1} \; \; ...(i)
2x_{1}+4X_{2}=b_{2}\; \; ...(ii)
3x_{1}+7x_{2}=b_{3}\; \; ...(iii)
3x_{1}+9x_{2}=b_{4}\; \; ...(iv)
From equations (ii) and (i)
We can write,
b_{2}=2[x_{1}+2x_{2}]=2b_{1}
From option (C):
3b_{1}-6b_{3}+b_{4}=3[x_{1}+2x_{2}]-6[3x_{1}+7x_{2}]+3x_{1}+9x_{2}\neq 0
From option (A):
b_{2}=2b_{1}
and b_{1}-3b_{3}+b_{4}=6[x_{1}+2x_{2}]-3[3x_{1}+7x_{2}]+[3x_{1}+9x_{2}]=0
6b_{1}-3b_{3}+b_{4}=0
Hence, answer is option (A).
Question 6
The number of distinct eigenvalues of the matrix
\begin{bmatrix} 2 & 2 & 3 &3 \\ 0& 1 & 1 & 1\\ 0 & 0 & 3 &3 \\ 0&0 & 0 &2 \end{bmatrix}
is equal to ____
A
1
B
2
C
3
D
4
GATE EC 2019   Engineering Mathematics
Question 6 Explanation: 
A=\left[\begin{array}{llll} 2 & 2 & 3 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 2 \end{array}\right]
Eigen values are 2,1,3,2
Distinct eigen values are 2,1,3[/latex]
\therefore Number of distinct eigen values =3
Question 7
Consider matrix A=\begin{bmatrix} k & 2k\\ k^{2}-k & k^{2} \end{bmatrix} and vector x=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}. The number of distinct real values of k for which the equation Ax=0 has infinitely many solutions is _______.
A
1
B
2
C
3
D
4
GATE EC 2018   Engineering Mathematics
Question 7 Explanation: 
A X=0 has infinitely many solutions
\begin{aligned} \text{So, }\quad|A|=0 ;&\left|\begin{array}{cc}k & 2 k \\ k^{2}-k k^{2}\end{array}\right|=0 \\ k^{3}-2 k^{3}+2 k^{2} &=0 ; \quad k^{2}(2-k)=0 \\ k &=0,2 \end{aligned}
\Rightarrow "two" distinct values of k
Question 8
Let M be a real 4\times4 matrix. Consider the following statements:
S1: M has 4 linearly independent eigenvectors.
S2: M has 4 distinct eigenvalues.
S3: M is non-singular (invertible).
Which one among the following is TRUE?
A
S1 implies S2
B
S1 implies S3
C
S2 implies S1
D
S3 implies S2
GATE EC 2018   Engineering Mathematics
Question 8 Explanation: 
Eigen vectors corresponding to distinct eigen values are linearly independent.
Hence, "S2 implies S1 ".
Question 9
The rank of the matrix \begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ -1 & 0& 0& 0 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix} is ___________.
A
4
B
5
C
6
D
3
GATE EC 2017-SET-2   Engineering Mathematics
Question 9 Explanation: 
A=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ -1 & 0& 0& 0 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix}
R_4\rightarrow R_4+R_1
=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ 0 & -1& 0& 0 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix}
R_4\rightarrow R_4+R_3
=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ 0 & 0& -1& 0 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix}
R_4\rightarrow R_4+R_2
=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ 0 & 0& 0& -1 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix}
R_4\leftrightarrow R_3 \text{ and }R_5\rightarrow R_5+R_4
=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 1& -1 & &0 \\ 0 & 0 &1 & -1&0 \\ 0 & 0& 0& -1 &1 \\ 0 & 0 & 0 &0 &0 \end{bmatrix}
From here,
\therefore \;\; \rho (A)=4
Question 10
The rank of the matrix M=\begin{bmatrix} 5 & 10 &10 \\ 1& 0 &2 \\ 3 & 6&6 \end{bmatrix} is
A
0
B
1
C
2
D
3
GATE EC 2017-SET-1   Engineering Mathematics
Question 10 Explanation: 
\begin{aligned} &M=\left[\begin{array}{lll} 5 & 10 & 10 \\ 1 & 0 & 2 \\ 3 & 6 & 6 \end{array}\right]\\ &R_{1} \leftrightarrow R_{2}:\\ &\left[\begin{array}{lll} 1 & 0 & 2 \\ 5 & 10 & 10 \\ 3 & 6 & 6 \end{array}\right]\\ &R_{2} \leftarrow R_{2}-5 R_{1} \text { and } R_{3} \leftarrow R_{3}-3 R_{1}\\ &\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 10 & 0 \\ 0 & 6 & 0 \end{array}\right]\\ &R_{3} \leftarrow R_{3}-\frac{6}{10} R_{2}\\ &\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 10 & 0 \\ 0 & 0 & 0 \end{array}\right] \end{aligned}
Which is in Echelon form
Rank of matrix M is,
\rho (M) = 2
There are 10 questions to complete.