# 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 $b_{1}-3b_{3}+b_{4}=6[x_{1}+2x_{2}]-3[3x_{1}+7x_{2}]+[3x_{1}+9x_{2}]=0$
$6b_{1}-3b_{3}+b_{4}=0$
 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$\times$4 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.