The matrix $M = \left[ {\,\begin{array}{*{20}{c}} a&4&5&6\\\7&a&4&5\\\8&7&a&4\\\9&8&7&a \end{array}\,} \right]$ is a ______ matrix.

• A) Toeplitz
• B) Rectangular
• C) Identity
• D) Diagonal

Two matrices with the same characteristic polynomial _____________ similar.

• A) may not be
• B) must be
• C)
• D)

If the real part of the eigenvalue is zero, then the trajectories form ------------- around the origin.

• A) Parabola
• B) None of these
• C) Ellipse
• D) Hyperbola

A transformation $T:{R^n} \to {R^m}$ is a rule that assigns to each vector x in ${R^n}$ , an image vector T(x) in ${R^m}$. The set ${R^n}$ is called the__________ , and ${R^m}$ is called the _______.

• A) co-domain of T, domain of T
• B) domain of T, co-domain of T
• C)
• D)

When a matrix A has a complex eigenvalue with positive real part, the trajectories spiral --------------

• A) Inwards
• B) Outwards
• C)
• D)

Eigenvalues of a square matrix are always

• A) Real and imaginary
• B) Positive.
• C) Real
• D) Negative

If n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).

• A) True
• B) False
• C)
• D)

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

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

If $A$ is invertible and $b$ in $R^n$ be any vector. Then, we must have a matrix ${A^{ - 1}}b$ ,which is a solution of _______.

• A) $${A^2}x = b$$
• B) $${A^{ - 1}}x = b$$
• C) $$Ax = b$$
• D) $${A^t}x = b$$

Let A be n x n matrix, then A is invertible if and only if

• A) det A is not zero
• B) det A is zero
• C)
• D)

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

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

An [Math Processing Error]n×n$n \times n$ matrix is diagonalizable with n --------- eigenvalues

• A) 0
• B) Distinct
• C) Similar
• D) Identical

If the augmented matrices of two linear systems are row equivalent, then the two systems have the ______ solution set.

• A) inconsistant
• B) same
• C) different
• D) trivial

For u, v vectors in ${R^n}$, the distance between u and v is written as

• A) None of these
• B) $dist(u,v) = ||u - v||$
• C) $dist(u,v) = ||u - u||$
• D) $dist(u,v) = ||v-v||$

A unit vector is obtained by ------

• A) $u={||v||}/{||v||}
• B) $u=v/{||v||}
• C) None of the above
• D) $u={||v||}/v

The equation $b=A\mathop x\limits^{^}+(b-A\mathop x\limits^{^}$ is a decomposition of $b$ into the sum of a vector in ------------ and a vector orthogonal to $ColA$.

• A) $RowA$
• B) $ColA$
• C)
• D)

If x is orthogonal to both u and v, then x must be _______ to u – v.

• A) equal
• B) orthogonal
• C) orthonormal
• D) parallel

The matrix $A^{T}A$ is invertible iff the columns of $A$ are

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

For two vectors to be orthonormal, the vectors are also said to be orthogonal. The reverse of the same

• A) Is not true
• B) Is true
• C)
• D)

The two vectors are said to be equivalent if

• A) None of the above
• B) Same length
• C) Same direction
• D) Both (a) and (b)