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.