The set of all solutions of $(A-\lambdaI)x$ is just the ……………………. Of the matrix $A$.

• A) Topological space
• B) Metric space
• C) None of the above
• D) Null space

The act of changing A into P^-1 AP is called a similarity transformation

• A) False
• B) True
• C)
• D)

The eigenvalues of a 4 by 4 matrix [A] are given as 2, -3, 1, 3 and 7. The $|det(A)|$ then is

• A) 530
• B) cannot be determined
• C) 546
• D) 25

The set $B={1,x,x^2,…,x^n}$ forms a basis for the vector space $P_n$ of polynomial of degree

• A) $\leq n$
• B) $n$
• C) $\geq n$
• D) None of these

Rank of the matrix A =\[\left[ {\begin{array}{*{20}{c}} 0&0&0&0 \\\ 4&6&1&0 \\\ 1&0&0&0 \end{array}} \right]\]

• A) 0
• B) 3
• C) 2
• D) 1

The point of intersection of axes in three dimension coordinates system is called?

• A) Ordinate
• B) Parallel pair order
• C) Collective coordinate
• D) Collective intercept

An n × n real matrix A is invertible if and only if the span of the rows of A is R^n

• A) True
• B) False
• C)
• D)

If $\lambda$ is an eigenvector of $A$, then every nonzero vector $x$ such that $Ax=\lambda x$ is called an ----------------- of $A$ corresponding to -----------

• A) Eigenvalue, $A$
• B) Eigenvector, $\lambda$
• C) Eigenvalue, $\lambda$
• D) Eigenvector, $A$

Every square matrix A has at least one ----------.

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

$Dim(M_{m\times n})$=-------

• A) 0
• B) $m-n$
• C) $m+n$
• D) $mn$*

The set of unit vectors ${i,j,k}$form a basis of

• A) $R^4$
• B) $R$
• C) $R^2$
• D) $R^3$

A coordinate system on a set consists of a ______________ mapping of the points in the set into R^n

• A) One-to-one
• B) Onto
• C)
• D)

If $B=/{b_{1}, b_{2}, …, b_{n}/}$ and $E$ is the standard basis $/{e_{1}, e_{2}, …, e_{n}/}$ in $R^{n}$, then

• A) $[b_{1}]_{E}=b_{1}$
• B) $[b_{1}]_{E}=0$
• C)
• D)

If vector space is not spanned by a finite set of vectors, then V is said to be --------

• A) Finite-dimensional
• B) Infinite-dimensional
• C)
• D)

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) ${A^{( - 1)}} = B,{B^{( - 1)}} = A,$