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

$$\left( {\begin{array}{*{20}{c}} 3&0&0 \\\ 5&4&0 \\\ 3&6&1 \end{array}} \right)$$ The Eigen-values are

• A) (3,4,2)
• B) (3,4,5)
• C) (2.3,5)
• D) (3,4,1)

The Invertible Matrix Theorem applies only to ______ matrices.

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

Which of the following is (are) the Eignvalue(s) of the matrix? $$A = \left[ {\,\begin{array}{*{20}{c}} 3&0\\\5&3 \end{array}\,} \right]$$

• A) 3, 5
• B) 5
• C) 3
• D) 3, 4

If the inverse of a linear transformation exits then it is ______.

• A) non-invertible
• B) nonlinear
• C) Not unique
• D) Unique

The trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. Its Eigen values are

• A) -37 and -1
• B) -7 and 5.
• C) -30 and -5
• D) 17.5 and -2

The characteristics polynomial of a 3x 3 identity matrix is __________, if x is the eigen values of the given 3 x 3 identity matrix.

• A) (1 - x)^3
• B) (x - 1)^3
• C) X^3
• D) (x + 1)^3

Eigenvector(s) of the matrix \[\left[ {\begin{array}{*{20}{c}}
0&0&a \\
0&0&0 \\
0&0&0
\end{array}} \right]\] is (are):

• A) None of the above
• B) (0, 0, 1)
• C) (0, 0, a)
• D) (0, a, 0)

A square matrix A is said to be diagonal if A is similar to a matrix

• A) Column matrix
• B) Diagonal matrix
• C) Zero matrix
• D) None of the above

A matrix $A$ is diagonalizable if and only if there are enough eigenvectors to form a basis of

• A) None of the above
• B) $R^n$
• C) $R^2$
• D) $R^3$

Multiplication of a partitioned matrix by a scalar is also computed ______.

• A) Row by row
• B) Block by block
• C) Diagonal by diagonal
• D) Column by column

Each pair of eigenvalue and its corresponding eigenvectors provides a solution of the equation $$x' = Ax$$ which is called

• A) eigenfunctions
• B) vector space
• C) solution matrix
• D) dynamical system

If x+2 is a factor of the characteristic polynomial of matrix C then an eigenvalue of C is

• A) -2
• B) 0
• C) 2
• D) 1/2

An n \cross n matrix with ------distinct eigenvalues is diagonalizable.

• A) $n+1$
• B) $n+2$
• C) $n$
• D) none of the above

What is the maximum possible number of pivots in a 6x6 matrix?

• A) 6
• B) 2
• C) 4
• D) 0