The complex conjugate of a complex vectorx in C^n is the vector \bar{x} in C^{n} whose entries are the ------------ conjugates of the entries in x

• A) Complex
• B) Real
• 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) Eigenvector, A
• B) Eigenvalue, \lambda
• C) Eigenvalue, A
• D) Eigenvector, \lambda

\begin{gathered} {x_1} + 2{x_2} + 3{x_3} = 7 \hfill \\\\ 4{x_1} + {x_2} + 2{x_3} = 2 \hfill \\\\ - 4{x_1} + 3{x_2} + 9{x_3} = 4 \hfill \\\\\\ \end{gathered} The augmented matrix for the system is -----------

• A) \left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ { - 4}&1&2 \\\\\\ 4&3&9 \end{array}} \right]
• B) \left( {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ 4&1&2 \\\\\\ { - 4}&3&9 \end{array}} \right)
• C) \left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ { - 4}&1&2 \\\\\\ 4&3&9 \end{array}\,\,\,\,\begin{array}{*{20}{c}} 7 \\\\\\ 2 \\\\\\ 4 \end{array}} \right]
• D) \left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ 4&1&2 \\\\\\ { - 4}&3&9 \end{array}\,\,\,\,\begin{array}{*{20}{c}} 7 \\\\\\ 2 \\\\\\ 4 \end{array}} \right]

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

\phi is the angle between the positive x-axis and the ray

• A) From (0,0) through (a,b)
• B) From (a,b) through (0,0)
• C) From (1,1) through (0,0)
• D) None of these

If A is an invertible square matrix then

• A) None of the above
• B) {\left( {{A^T}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T}
• C) {\left( {{A^T}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^{-1}}
• D) {\left( {{A^T}} \right)^T} = {\left( {{A^{ - 1}}} \right)^T}

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^3
• D) R^2

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

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

If \lambda is an eigenvalue of a matrix A and x is a corresponding eigenvector, and if k is any positive integer, then {\lambda ^k} is an eigenvalue of ______ and x is a corresponding eigenvector.

• A) \lambda ^k
• B) A^k
• C) \lambda
• D) A

Let A be a real 2 by 2 matrix with complex eigen values \lambda = a - bi(b \ne 0) and associated eigenvectors v in {C^2} , then

• A) A = {P^{ - 1}}CP
• B) A = PC{P^{ - 1}}
• C) None of the above
• D) P = AC{P^{ - 1}}

If \lambda + 7 is a factor of the characteristic polynomial of a matrix C , then which of the following is the Eigenvalue of C ?

• A) 0
• B) \frac{1}{7}
• C) 7
• D) -7

Diagonalization is a process of transforming a vector A to the form

• A) A=PDP^{-1}
• B) A=P^{-1}DP
• 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)

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, 5
• C) 3
• D) 3, 4

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)

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) 25
• C) 546
• D) cannot be determined

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

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

Let A be an n \times n matrix whose distinct eigen values are \lambda_1, \lambda_2, …, \lambda_p. The matrix A is diagonalizable if and only if the ----------- of the dimensions of the distinct eigen spaces is equal to n

• A) sum
• B) Product
• C)
• D)

\[\begin{gathered} {x_1} + 2{x_2} + 3{x_3} = 7 \hfill \\\\ 4{x_1} + {x_2} + 2{x_3} = 2 \hfill \\\\ - 4{x_1} + 3{x_2} + 9{x_3} = 4 \hfill \\\\\\ \end{gathered} \] The augmented matrix for the system is -----------

• A) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ 4&1&2 \\\\\\ { - 4}&3&9 \end{array}\,\,\,\,\begin{array}{*{20}{c}} 7 \\\\\\ 2 \\\\\\ 4 \end{array}} \right]$
• B) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ { - 4}&1&2 \\\\\\ 4&3&9 \end{array}} \right]$
• C) $\left[ {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ { - 4}&1&2 \\\\\\ 4&3&9 \end{array}\,\,\,\,\begin{array}{*{20}{c}} 7 \\\\\\ 2 \\\\\\ 4 \end{array}} \right]$
• D) \[\left( {\begin{array}{*{20}{c}} 1&2&3 \\\\\\ 4&1&2 \\\\\\ { - 4}&3&9 \end{array}} \right)\]

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

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