Eigen Values and Eigen Vectors

Q. In the given matrix $\left[\begin{array}{ccc}1& -1& 2\\ 0& 1& 0\\ 1& 2& 1\end{array}\right]$, one of the eigen value is 1. The eigen vectors corresponding to the eigen value 1 are

1. $\alpha (4,2,1)|\alpha \ne 0,\alpha ϵR$
2. $\alpha (-4,2,1)|\alpha \ne 0,\alpha ϵR$
3. $\alpha (\sqrt{2},0,1)|\alpha \ne 0,\alpha ϵR$
4. $\alpha (-\sqrt{2},2,1)|\alpha \ne 0,\alpha ϵR$

Q. For the matrix $\left[\begin{array}{cc}4& 2\\ 2& 4\end{array}\right]$, the eigen value corresponding to the eigen vector $\left[\begin{array}{c}101\\ 101\end{array}\right]$ is

1. $2$
2. $4$
3. $6$
4. $8$

Q.

$(a.i)i+(a.j)j+(a.k)k$ is equal to?

1. $a$

2. $2a$

3. $3a$

4. $0$

Q. The eigen values of the matrix given below are: $\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& -3& -4\end{array}\right]$

1. (0, -1, -3)
2. (0, -2, -3)
3. (0, 2, 3)
4. (0, 1, 3)

Q. The eigen values of the matrix $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ are

1. i and -i
2. 1 and -1
3. 0 and 1
4. 0 and -1

Q. The minimum eigen value of the following matrix is $$\left[\begin{array}{ccc}3& 5& 2\\ 5& 12& 7\\ 2& 7& 5\end{array}\right]$$

1. 0
2. 1
3. 2
4. 3

Q. Let P = $\left[\begin{array}{cc}3& 1\\ 1& 3\end{array}\right]$ Consider the set S of all vectors $\left[\begin{array}{c}x\\ y\end{array}\right]$ such that $a^2+b^2$ = 1 where $\left(\begin{array}{c}a\\ b\end{array}\right)$ = P$\left(\begin{array}{c}x\\ y\end{array}\right)$ then s is

1. A circle of radius $\sqrt{10}$
2. A circle of radius $\frac{1}{\sqrt{10}}$
3. An ellipse with major axis along $\left[\begin{array}{c}1\\ 1\end{array}\right]$
4. An ellipse with minor axis along $\left[\begin{array}{c}1\\ 1\end{array}\right]$

Q. Consider the 5 x 5 matrix $A=\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 5& 1& 2& 3& 4\\ 4& 5& 1& 2& 3\\ 3& 4& 5& 1& 2\\ 2& 3& 4& 5& 1\end{array}\right]$

It is given that A has only one real eigen value. Then the real eigen value A is

1. -25
2. 0
3. 15
4. 25

Q. The determinant of a 2 x 2 matrix is 50. If one eigen value of the matrix is10, the other eigen value is

Q. The eigen values of the matrix $\left[\begin{array}{cccc}2& -1& 0& 0\\ 0& 3& 0& 0\\ 0& 0& -2& 0\\ 0& 0& -1& 4\end{array}\right]$ are

1. $2,-2,1,-1$
2. $2,3,-2,4$
3. $2,3,1,4$
4. None

Q. The Eigen vectors of the matrix
$\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$ is/are

1. $(1,0)$
2. $(0,1)$
3. $(1,1)$
4. $(1,-1)$

Q. If A is $3\times 3$ matrix and Trace A = 9, |A| = 24 and one of the eigen values is 3, then sum of other eigen values is __________.

1. 5
2. 8
3. 6
4. 9

Q. The eigen values of the matrix
$\left[\begin{array}{cc}5& 3\\ 3& -3\end{array}\right]$ are:

1. $6$
2. $5$
3. $-3$
4. $-4$

Q. Consider the matrix P = $\left[\begin{array}{ccc}\frac{1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}\\ 0& 1& 0\\ \frac{-1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}\end{array}\right]$ which one of the following statements about P is INCORRECT?

1. Determinant of P is equal to 1.
2. P is orthogonal
3. Inverse of P is equal to its transpose.
4. All eigen values of P are real numbers

Q. A system matrix is given as follows :
$A=\left[\begin{array}{ccc}0& 1& -1\\ -6& -11& 6\\ -6& -11& 5\end{array}\right]$
The absolute value of the ratio of the maximum eigen value to the minimum eigen value is __

1. 0.333

Q. The lowest eigen value of the 2 x 2 matrix $\left[\begin{array}{cc}4& 2\\ 1& 3\end{array}\right]$ is .

1. 2

Q. Consider the matrix:

P = $\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$

The number of distinct eigenvalues of P is:

1. 3
2. 0
3. 1
4. 2

Q. An eigen vector of $P=\left[\begin{array}{ccc}1& 1& 0\\ 0& 2& 2\\ 0& 0& 3\end{array}\right]$ is

1. ${\left[\begin{array}{ccc}-1& 1& 1\end{array}\right]}^T$
2. ${\left[\begin{array}{ccc}1& 2& 1\end{array}\right]}^T$
3. ${\left[\begin{array}{ccc}1& -1& 2\end{array}\right]}^T$
4. ${\left[\begin{array}{ccc}2& 1& -1\end{array}\right]}^T$

Q. The eigen values of the matrix A = $\left[\begin{array}{ccc}1& -1& 5\\ 0& 5& 6\\ 0& -6& 5\end{array}\right]$ are

1. 1, 5, 6
2. 1, -5, $\pm$j6
3. 1, 5$\pm$j6
4. 1, 5, 5

Q.

What is the meaning of determinants?

Q. The eigen values of matrix $\left[\begin{array}{cc}9& 5\\ 5& 8\end{array}\right]$ are

1. -2.42 and 6.86
2. 3.48 and 13.53
3. 4.70 and 6.86
4. 6.86 and 9.50

Q. Consider a 2 x 2 square matrix A = $\left[\begin{array}{cc}\sigma & X\\ \omega & \sigma \end{array}\right]$

Where x is unknown. If the eigen values of the matrix A are ($\sigma +j\omega$) and ($\sigma -j\omega$), then x is equal to

1. +j$\omega$
2. -j$\omega$
3. +$\omega$
4. -$\omega$

Q. [1, 1, 2] is an eigen vector of the matrix, $A=\left[\begin{array}{ccc}3& 1& -1\\ 2& 2& -1\\ 2& 2& 0\end{array}\right]$ corresponding to the eigen value x. The value of x is _______ .

1. 2

Q. The vector $\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]$ is an eigen vector of $A=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& -6\\ -1& -2& 0\end{array}\right]$. The corresponding eigen value of $A$ is _____.

1. $1$
2. $2$
3. $5$
4. $-1$

Q. The matrix $\left[\begin{array}{cc}2& -4\\ 4& -2\end{array}\right]$ has

1. Real eigen values and eigen vectors
2. Real eigen values but complex eigen vectors
3. Complex eigen values but real eigen vectors
4. Complex eigen values and eigen vectors

Q. In matrix equation [A] {X} = {R},

[A] = $\left[\begin{array}{ccc}4& 8& 4\\ 8& 16& -4\\ 4& -4& 15\end{array}\right]$, $\left\{X\right\}=\left\{\begin{array}{c}2\\ 1\\ 4\end{array}\right\}$ and $\left\{R\right\}=\left\{\begin{array}{c}32\\ 16\\ 64\end{array}\right\}$

One of the eigen values of matrix [A] is

1. 16
2. 8
3. 15
4. 4

Q. Let the Eigen vector of the matrix $\left[\begin{array}{cc}1& 2\\ 0& 2\end{array}\right]$ be written in the form $\left[\begin{array}{c}1\\ a\end{array}\right]$ and $\left[\begin{array}{c}1\\ b\end{array}\right]$. What is the value of (a + b)?

1. 0
2. $\frac{1}{2}$
3. 1
4. 2

Q. A 2 x 2 matrix M has eigen values 2 & 3 with eigen vectors $\left[\begin{array}{c}2\\ 1\end{array}\right]\&\left[\begin{array}{c}1\\ 2\end{array}\right]$ respectively. The sum of all elements of Matrix M is _______.

1. 5

Q. M is a $2\times 2$ matrix with eigen values 2 and 5.The eigen values of $M^{-1}$are

1. -2 and -5
2. 2 and 5
3. 0.5 and 0.2
4. Can't be determined

Q. One of the eigen value for the following matrix is

$\left[\begin{array}{cc}x& 2\\ 8& x\end{array}\right]$

1. x - 4
2. - x - 4
3. 4
4. -4