Le a matrix A has both negative and positive eigen values, so in this case origin behaves as a ----------- point

• A) Saddle
• B) Critical
• C)
• D)

Let $A$ be an $n by n$ matrix whose distinct eigen values are $\lambda_{1}, \lambda_{2},…, \lambda_{p}$. For $1 \leq k \leq p$, the dimension of the eigen space for $\lambda_{k}$ is -------- than or ------- to the multiplicity of the eigen value $\lambda_{k}$.

• A) Greater, equal
• B) Less, equal
• C)
• D)

If $\lambda - 2$ is a factor of the characteristic polynomial of matrix $A$ , then which of the following is the Eigenvalue of $A$ ?

• A) 2
• B) -2
• C) $\frac{1}{2}$
• D) 0

If a vector $x_0$ is specified, then the initial value problem is to construct the unique function $x$ such that --------

• A) $x’=Ax’$ and $x(0)=x_0$
• B) $x’=Ax$ and $x(0)=x_0$
• C)
• D)

What is Eigen value?

• A) A vector obtained from the coordinates
• B) A matrix determined from the algebraic equations
• C) It is the inverse of the transform
• D) A scalar associated with a given linear transformation

$${\rm{If}}\,{\rm{AB = I = BA \,\ for }}\,\ matrices{\rm{ }}A,\,B\;and\,I,{\rm{ }}where\;\,I\,is\,\ an \,\ identity \,\ matrix, \,\ then$$

• A) $B$ is inverse of $A$,
• B) All of the above.
• C) $A$ is inverse of $B$,
• D) ${A^{( - 1)}} = B,{B^{( - 1)}} = A,$

Electric circuit rotation is caused by sine and cosine functions if there eigenvalues are complex and hence the origin is called --------

• A) Spiral point
• B) Critical point
• C)
• D)

Let A be a real 2 by 2 matrix with complex eigen values $\lambda = a - bi(b \ne 0)$ and associated eigenvectors v in ${C^2}$ , then

• A) $A = PC{P^{ - 1}}$
• B) $A = {P^{ - 1}}CP$
• C) $P = AC{P^{ - 1}}$
• D) None of the above

$\phi$ is the angle between the positive x-axis and the ray

• A) None of these
• B) From (1,1) through (0,0)
• C) From (a,b) through (0,0)
• D) From (0,0) through (a,b)

Which one of the following statements is true for all real symmetric matrices?

• A) All the eigenvalues are positive.
• B) All the eigenvalues are distinct
• C) Sum of all the eigenvalues is zero.
• D) All the eigenvalues are real.

A blocked matrix in which blocks (blocked matrices) are repeated down the diagonals of the matrix is called a ______ matrix.

• A) Blocked identity
• B) Blocked rectangular
• C) Blocked diagonal-constant
• D) Blocked square

An n x n matrix A is diagonalizable if and only if A has ____ linearly independent eigenvectors.

• A) n - 1
• B) n
• C) n x n
• D) n + 1

A column replacement operation on A does not change the _____________.

• A) determinant
• B) matrix
• C) row
• D) column

The null-space of a linear transformation T is the set of all vectors v such that T (v) = ________.

• A) positive
• B) 0
• C) negative
• D) non-zero

If u + v =u + w, then:

• A) None of the above
• B) v ≠ w
• C) v + w.
• D) v = w.

Factorization is used to analyze and decouple --------

• A) Dynamical system
• B) Linear system
• C)
• D)

Which of the following is true for the matrix $$M = \left[ \begin{array}{l} \,{M_{\,11}}\,\,\,\,\,\,\,{M_{\,12}}\,\,\,\,\,\,\,{M_{\,13}}\\\ O\,\,\,\,\,\,\,\,\,\,\,\,{M_{\,22}}\,\,\,\,\,\,\,{M_{\,23}}\\\ O\,\,\,\,\,\,\,\,\,\,\,\,\,O\,\,\,\,\,\,\,\,\,\,\,\,{M_{\,33}}\, \end{array} \right]\,;$$ where ${M_{\,\,11}}$ , ${M_{\,22}}$ and ${M_{\,33}}$ are square sub-matrices , and $O$ is a zero sub-matrix ?

• A) It is a block lower triangular matrix.
• B) It is diagonal-constant matrix.
• C) It is a Null matrix.
• D) It is a block upper triangular matrix.

An $n \times n$ matrix $A$ is said to be diagonalizable if and only if $A$ has $n$ ------------------------------ eigenvectors.

• A) Linearly dependent
• B) Linearly Independent
• C)
• D)

Let $A$ be an $n \times n$ matrix whose distinct eigen values are $\lambda_1, \lambda_2, …, \lambda_p$. The matrix $A$ is diagonalizable if and only if the ----------- of the dimensions of the distinct eigen spaces is equal to $n$.

• A) sum
• B) Product
• C)
• D)

Let V be a one Eigen vector then conjugate eigen vector is represented by,

• A) $\mathop V\limits^ \sim$
• B) $V'$
• C) V
• D) $\mathop V\limits^{\_\_}$