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) $\mathop V\limits^{\_\_}$

Two equivalent vectors must have the same initial point:

• A) None of the above
• B) May be.
• C) True
• D) False

If $A$ be $n \times n$ matrix, then $det (A^T) =$ ______________.

• A) det A
• B) inverse of A
• C) $$det A^T$$
• D) 1 / det A

Diagonalization is a process of transforming a vector $A$ to the form

• A) $A=PDP^{-1}$
• B) $A=P^{-1}DP$
• C)
• D)

A 3 x 3 identity matrix have three and __________eigen values.

• A) distinct
• B) same
• C)
• D)

The complex conjugate of a complex vector$x$ in $C^n$ is the vector $\bar{x}$ in $C^{n}$ whose entries are the ------------ conjugates of the entries in $x$.

• A) Real
• B) Complex
• C)
• D)

Let $A$ and $B$ be the square matrices. Then, $A$ and $B$ are invertible with $B = {A^{ - 1}}$ and $A = {B^{ - 1}}$ if and only if $AB = BA$ equals to a (an) _____ matrix.

• A) Rectangular
• B) Identity
• C) Square
• D) Singular