For a given matrix \left[ {\begin{array}{*{20}{c}} 1&{ - 2}&{ - 3}&{ - 4}&{ - 4}&6 \\\ 0&0&1&2&1&{ - 4} \\\ 0&0&0&0&1&7 \\\ 0&0&0&0&0&0 \end{array}} \right] which of the following are basis vectors.

• A) \left[ {\begin{array}{*{20}{c}} 1 \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 2} \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 3} \\\ 1 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} 6 \\\ { - 4} \\\ 7 \\\ 0 \end{array}} \right]
• B) \left[ {\begin{array}{*{20}{c}} 1 \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 2} \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 3} \\\ 1 \\\ 0 \\\ 0 \end{array}} \right]
• C) \left[ {\begin{array}{*{20}{c}} 1 \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 3} \\\ 1 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 4} \\\ 1 \\\ 1 \\\ 0 \end{array}} \right]
• D) \begin{gathered} \left[ {\begin{array}{*{20}{c}} 1&{ - 2}&{ - 3}&{ - 4}&{ - 4}&6 \end{array}} \right] \hfill \\ \left[ {\begin{array}{*{20}{c}} 1&0&1&2&1&{ - 4} \end{array}} \right] \hfill \\ \left[ {\begin{array}{*{20}{c}} 0&0&0&0&1&7 \end{array}} \right] \hfill \\\ \end{gathered}

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}=0
• B) [b_{1}]_{E}=b_{1}
• C)
• D)

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

• A) false
• B) True
• C)
• D)

The pivot columns of a matrix A form a basis for -----------

• A) ColA
• B) RowA
• C)
• D)

If~ A ~is~ an~ m \times n ~matrix ~then

• A) rank A + \dim NulA = n
• B) rank A = n
• C) rank A = \dim NulA + n
• D) rank A = \dim Nul A

Rank of the matrix A=

• A) 1
• B) 3
• C) 2
• D) 0

For a non-singular matrix A, A^{-1} is equal to------

• A) Adj(A)/det(A)*
• B) None of the above.
• C) det(A)*A
• D) det(A)*Adj(A)

\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)

An n \times n matrix A is said to be diagonalizable if and only if A has n ------------------------------ eigenvectors.

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

An n \times n matrix is diagonalizable with n --------- eigenvalues

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

\[{\rm{If}}\,{\rm{AB = I = BA \,\ for }}\,\ matrices{\rm{ }}A,\,B\;and\,I,{\rm{ }}where\;\,I\,is\,\ an \,\ identity \,\ matrix, \,\ then\]

• A) ${A^{( - 1)}} = B,{B^{( - 1)}} = A,$
• B) All of the above.
• C) $B$ is inverse of $A$,
• D) $A$ is inverse of $B$,

{\rm{If}}\,{\rm{AB = I = BA \,\ for }}\,\ matrices{\rm{ }}A,\,B\;and\,I,{\rm{ }}where\;\,I\,is\,\ an \,\ identity \,\ matrix, \,\ then

• A) All of the above.
• B) {A^{( - 1)}} = B,{B^{( - 1)}} = A,
• C) A is inverse of B,
• D) B is inverse of A,

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

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

Let A be a real 2 by 2 matrix with complex eigen values \lambda=a-b+{i}, (b\neq 0) and associated eigenvectors v in C^2, then A=PCP^{-1}, where P=--------------

• A) P=[Rev Imv]
• B) P=[-Rev -Imv]
• C) P=[-Rev Imv]
• D) P=[Rev -Imv]

If 3 is an eigenvalue of A and x is a corresponding eigenvector, then what is the eigenvalue of A^2 ?

• A) 9
• B) 6
• C) 12
• D) 3

\lambda is an eigenvalue of a matrix A if and only if the equation (A-\lambdaI)x=0 has a ---------

• A) Non-trivial solution
• B) Trivial Solution
• C)
• D)

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) 5
• B) 3
• C) 3, 4
• D) 3, 5

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) Square
• C) Identity
• D) Singular

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

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

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