# Linear Algebra

 Question 1
The state equation of a second order system is

$\dot{x}(t)=A x(t), x(0)$ is the initial condition.

Suppose $\lambda_{1}$ and $\lambda_{2}$ are two distinct eigenvalues of $A$ and $v_{1}$ and $v_{2}$ are the corresponding eigenvectors. For constants $\alpha_{1}$ and $\alpha_{2}$, the solution, $x(t)$, of the state equation is
 A $\sum_{i=1}^{2} \alpha_{i} e^{\lambda_{i} t} v_{i}$ B $\sum_{i=1}^{2} \alpha_{i} e^{2 \lambda_{i} t} v_{i}$ C $\sum_{i=1}^{2} \alpha_{i} e^{3 \lambda_{i} t} v_{i}$ D $\sum_{i=1}^{2} \alpha_{i} e^{4 \lambda_{i} t} v_{i}$
GATE EC 2023   Engineering Mathematics
 Question 2
Let $x$ be an $n \times 1$ real column vector with length $l=\sqrt{x^{T} x}$. The trace of the matrix $P=x x^{T}$ is
 A $l^{2}$ B $\frac{l^{2}}{4}$ C $l$ D $\frac{l^{2}}{2}$
GATE EC 2023   Engineering Mathematics
Question 2 Explanation:
Given,
$l=\sqrt{x^{T} x}, P=\left(x x^{T}\right)_{n \times n}$

Let
\begin{aligned} (x)_{n \times 1} & =\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \\ \vdots \\ x_{n} \end{array}\right] \\ l & =\sqrt{x^{T} x}=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\ldots x_{n}^{2}} \\ P & =x x^{T} \\ &=\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \\ \cdot \\ x_{n} \end{array}\right]\left[x_{1} x_{2} x_{3} \ldots x_{n}\right]\\ P&=\left[ \begin{array}{lllll} x_{1}^{2} & & & & \\ & x_{1}^{2} & & & \\ & & - & & \\ & & & - & \\ & & & & x_{n}^{2} \end{array} \right] \end{aligned}
Trace of $P=x_{1}^{2}+x_{2}^{2}+ . . .+ x_{n}^{2}=l^2$

 Question 3
Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{v_{k} \mid 1 \leq k \leq n\right\}$, respectively. For any invertible matrix $P$, the sets of eigenvalues and eigenvectors of the matrix $A$, where $B=P^{-1} A B$, respectively, are
 A $\left\{\lambda_{k} \operatorname{det}\mid 1 \leq k \leq n\right\}$ and $\left\{P v_{k} \mid 1 \leq k \leq n\right\}$ B $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{v_{k} \mid 1 \leq k \leq n\right\}$ C $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{P v_{k} \mid 1 \leq k \leq n\right\}$ D $\left\{\lambda_{k} \mid 1 \leq k \leq n\right\}$ and $\left\{P^{-1} v_{k} \mid 1 \leq k \leq n\right\}$
GATE EC 2023   Engineering Mathematics
Question 3 Explanation:
\begin{aligned} & B & =P^{-1} A P \\ A & =P B P^{-1}\end{aligned}

$\Rightarrow A, B$ are called matrices similar.
$\Rightarrow$ Both $A, B$ have same set 7 eigen values
But eigen vectors of $A, B$ are different.

Let $B X=\lambda X$
$\Rightarrow \quad\left(P^{-1} A P\right) X=\lambda X$
$\Rightarrow \quad A(P X)=\lambda(P X)$

$\therefore$ Eigen vectors of $A$ are $P X$.
 Question 4
The rate of increase, of a scalar field $f(x, y, z)=x y z$ in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is
 A $\frac{2}{3}$ B $\frac{4}{3}$ C $2$ D $4$
GATE EC 2023   Engineering Mathematics
Question 4 Explanation:
\begin{aligned} f(x, y, z) & =x y z \\ \overline{\nabla f} & =\hat{i} f_{x}+\hat{j} f_{y}+\hat{k} f_{z} \\ & =\hat{i}(y z)+\hat{j}(x z)+\hat{k}(x y) \\ \overline{\nabla f}_{(0,2,1)} & =\hat{i}(2)+0 \hat{j}+0 \hat{k} \end{aligned}

Directional derivative,
\begin{aligned} D \cdot D & =\overline{\nabla f} \cdot \frac{\bar{a}}{|\bar{a}|} \\ & =(2 \hat{i}+0 \hat{j}+0 \hat{k}) \cdot \frac{(2 \hat{i}+\hat{j}+2 \hat{k})}{\sqrt{2^{2}+1^{2}+2^{2}}}=\frac{4}{\sqrt{9}}=\frac{4}{3} \end{aligned}
 Question 5
Let $v_{1}=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right]$ and $v_{2}=\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right]$ be two vectors. The value of the coefficient $\alpha$ in the expression $v_{1}=\alpha v_{2}+e$, which minimizes the length of the error vector $e$, is
 A $\frac{7}{2}$ B $-\frac{2}{7}$ C $\frac{2}{7}$ D $-\frac{7}{2}$
GATE EC 2023   Engineering Mathematics
Question 5 Explanation:
\begin{aligned} e & =V_{1}-\alpha V_{2} \\ e & =(i+2 k+0 k)-\alpha(2 i+j+3 k) \\ \hat{e} & =(1-2 \alpha) \hat{i}+(2-\alpha) \hat{j}+(0-3 \alpha) \hat{k} \\ |\hat{e}| & =\sqrt{(1-2 \alpha)^{2}+(2-\alpha)^{2}+(-3 \alpha)^{2}} \\ |\hat{e}|^{2} & =5+14 \alpha^{2}-8 \alpha \text { to be minimum at } \frac{\partial e^{2}}{\partial \alpha}=28 \alpha-8=0 \\ \alpha & =\frac{2}{7} \text { stationary point } \end{aligned}

There are 5 questions to complete.