# Linear Algebra

 Question 1
Consider a matrix $A=\begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & -2\\ 0&1 &1 \end{bmatrix}$
The matrix $A$ satisfies the equation $6A^{-1}=A^2+cA+dI$ where $c$ and $d$ are scalars and $I$ is the identity matrix.
Then $(c+d)$ is equal to
 A 5 B 17 C -6 D 11
GATE EE 2022   Engineering Mathematics
Question 1 Explanation:
Characteristic equation:
\begin{aligned} |A-\lambda I|&=0 \\ \begin{vmatrix} 1-\lambda & 0 &0 \\ 0 & 4-\lambda & 2\\ 0 & -1 & 1-\lambda \end{vmatrix}&=0 \\ (1-\lambda )[(4-\lambda )(1-\lambda )+2] &=0\\ \lambda ^3-6\lambda ^2+11\lambda -6&=0 \end{aligned}
By cayley hamilton theorem
\begin{aligned} A^3-6A^2+11A-6 &=0 \\ A^2-6A+11I&=6A^{-1} \end{aligned}
On comparison : c= -6 and d = 11
Therefore, c + d = -6 + 11 = 5
 Question 2
$e^A$ denotes the exponential of a square matrix A. Suppose $\lambda$ is an eigenvalue and $v$ is the corresponding eigen-vector of matrix A.

Consider the following two statements:
Statement 1: $e^\lambda$ is an eigenvalue of $e^A$.
Statement 2: $v$is an eigen-vector of $e^A$.

Which one of the following options is correct?
 A Statement 1 is true and statement 2 is false. B Statement 1 is false and statement 2 is true C Both the statements are correct. D Both the statements are false.
GATE EE 2022   Engineering Mathematics
Question 2 Explanation:
Eigen value will change but eigen vector not change.

 Question 3
Consider a 3 x 3 matrix A whose (i,j)-th element, $a_{i,j}=(i-j)^3$. Then the matrix A will be
 A symmetric. B skew-symmetric. C unitary D null.
GATE EE 2022   Engineering Mathematics
Question 3 Explanation:
$for \; i=j\Rightarrow a_{ij}=(i-i)^3=0\forall i$
$for \; i\neq j\Rightarrow a_{ij}=(i-j)^3=(-(j-i))^3=-(j-i)^3=-a_{ji}$
$\therefore \; A_{3 \times 3 }$ is skew symmetric matrix.
 Question 4
Let A be a $10\times10$ matrix such that $A^{5}$ is a null matrix, and let I be the $10\times10$ identity matrix. The determinant of $\text{A+I}$ is ___________________.
 A 1 B 2 C 4 D 8
GATE EE 2021   Engineering Mathematics
Question 4 Explanation:
\begin{aligned} \text{Given}:\qquad A^{5}&=0\\ A x &=\lambda x \\ \Rightarrow \qquad \qquad A^{5} x &=\lambda^{5} x \quad(\because x \neq 0) \\ \Rightarrow \qquad \qquad \lambda^{5} &=0 \\ \Rightarrow \qquad \qquad \lambda &=0 \end{aligned}
Eigen values of $A+I$ given $\lambda+1$
$\because$ Eigen values of $I_{A}=1$
Hence $|A+I|=$ Product of eigen values $=1 \times 1 \times 1 \times \ldots 10$times
$=1$
 Question 5
Let p and q be real numbers such that $p^{2}+q^{2}=1$. The eigenvalues of the matrix $\begin{bmatrix} p & q\\ q& -p \end{bmatrix}$ are
 A 1 and 1 B 1 and -1 C j and -j D pq and -pq
GATE EE 2021   Engineering Mathematics
Question 5 Explanation:
Characteristic equation of A
\begin{aligned} \left|A_{2 \times 2}-\lambda I\right|&=(-1)^{2} \lambda^{2}+(-1)^{1} \text{Tr}(A) \lambda+|A|=0 \\ \lambda^{2}-(p-p) \lambda+\left(-p^{2}-q^{2}\right) &=0 \\ \Rightarrow \qquad \qquad\lambda^{2}-1 &=0 \\ \Rightarrow \qquad\qquad \lambda &=\pm 1 \end{aligned}

There are 5 questions to complete.