Rank of the matrix A=$$\left[ \begin{gathered} 0\,\,\,0\,\,\,0\,\,\,0 \hfill \\ 4\,\,\,6\,\,\,\,1\,\,\,\,0\, \hfill \\ 1\,\,\,\,\,0\,\,\,\,0\,\,\,\,0 \hfill \\\ \end{gathered} \right]$$

• A) 0
• B) 3
• C) 1
• D) 2

If A is a m×n matrix and $$A = A^T$$, which of the following must always be true?

• A) m = n
• B) m and n are different
• C)
• D)

In two coordinate system, ${\left( {\mathop P\limits_{C \leftarrow B} } \right)^{ - 1}}$ is the matrix that converts

• A) C-coordinates into C-coordinates
• B) B-coordinates into B-coordinates
• C) B-coordinates into C-coordinates
• D) C-coordinates into B-coordinates

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

• A) False
• B) True
• C)
• D)

The three dimensional coordinate system is one in which the coordinates intersect each other at

• A) Zero points
• B) Positive point
• C) Negative points
• D) Absolute point

The number of parameters in the solution set of Ax = 0 if A is a 5× 7 matrix of rank 3 then nullity (A)=?

• A) 5
• B) 4
• C) 7
• D) 3

If A is a triangular matrix then the eigenvalues of A are the entries on the main diagonal of A.

• A) True
• B) False
• C)
• D)

If Coordinates vector of the polynomial $$p = {a_0} + {a_1}x + {a_2}{x_2}$$ Relative to the basis S = $$\{ 1,x,{x^2}\} \,\,\,$$ for $${p_2}$$ then linear combination of the basis set S is

• A) $${[p]_s} = ({a_0} + {a_1} + {a_2})$$
• B) $${[p]_s} = [{a_0} + {a_1} + {a_2}]$$
• C)
• D)

Let $V$ be a five-dimensional vector space, and let $S$ be a subset of $V$ which spans $V$. Then $S$

• A) Must have infinitely many elements
• B) Must be linearly dependent
• C) Must have at most five elements
• D) Must be a basis for $V$

If two matrices A and B are row equivalent matrices, then which of the following is true

• A) a given set of column vectors of A is linearly dependent if and only if the corresponding column vectors of B are linearly dependent.
• B) a given set of column vectors of A is linearly independent if and only if the corresponding column vectors of B are exactly same.
• C) a given set of column vectors of A is linearly independent if and only if the corresponding column vectors of B are linearly independent.
• D) None of these is true

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

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

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

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

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

If one of the Eigenvalues of ${\left[ A \right]_{n \times n}}$ is zero, it implies

• A) The solution to [A][X]=[C] system of equations is unique
• B) The solution to [A][X]=[0]
• C) The determinant of [A] is nonzero
• D) The determinant of [A] is zero

$$\begin{gathered} if~ A ~is~ a~ 4 \times 5~ matrix~ with~ a~ two ~- {\text{dimentinal}}~ null space, ~ \hfill \\ what~ is~ rank ~of ~A \hfill \\\ \end{gathered}$$

• A) 3
• B) 6
• C) 5
• D) 7

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) 3
• B) 2
• C) 0
• D) 1

Rank of the matrix A=$$\left[ \begin{gathered} 0\,\,\,0\,\,\,0\,\,\,0 \hfill \\ 4\,\,\,6\,\,\,\,1\,\,\,\,0\, \hfill \\ 1\,\,\,\,\,0\,\,\,\,0\,\,\,\,0 \hfill \\\ \end{gathered} \right]$$

• A) 3
• B) 1
• C) 2
• D) 0

Let $V$ be a five-dimensional vector space, and let $S$ be a subset of $V$ which spans $V$. Then $S$

• A) Must have infinitely many elements
• B) Must have at most five elements
• C) Must be a basis for $V$
• D) Must be linearly dependent

If A is any matrix, then \[Rank(A) = Rank({A^T})\]

• A) True
• B) false
• C)
• D)

The vectors spaces $F(-\infty, +\infty)$ and $C(-\infty, +\infty)$ are ----------

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