MTH501- Linear Algebra Complete FinalTerm Unsolved Past Paper 2008

FINALTERM  EXAMINATION

Fall 2008

MTH501- Linear Algebra

Ref No: Time: 120 min

Marks: 70

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Question No: 1      ( Marks: 1 ) - Please choose one

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A matrix that results from applying a single elementary row operation to an identity matrix is called

       ► Invertible matrix

       ► Singular matrix

       ► Scalar matrix

       ► Elementary matrix

Question No: 3      ( Marks: 1 ) - Please choose one

For an n×n matrix (A^t)^t =

       ► A^t

       ► A

       ► A^-1

       ► (A^-1)^-1

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Question No: 4      ( Marks: 1 ) - Please choose one

What is the largest possible number of pivots a 4´6 matrix can have?

       ► 4

       ► 6

       ► 10

       ► 0

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Question No: 5      ( Marks: 1 ) - Please choose one

The characteristic polynomial of a 5´5 matrix is  ,the eigenvalues are

       ► 0,-5, 9

       ► 0,0,0,5,9

       ► 0,0,0,-5,9

       ► 0,0,5,-9

Question No: 6      ( Marks: 1 ) - Please choose one

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Find the characteristic equation of the given matrix

               

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       ►

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       ►

Question No: 7      ( Marks: 1 ) - Please choose one

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A is diagonalizable if  Where

       ► D is any matrix and P is an invertible matrix

       ► D is a diagonal matrix and P is any matrix

       ► D is a diagonal matrix and P is invertible matrix

       ► D is a invertible matrix and P is any matrix

Question No: 8      ( Marks: 1 ) - Please choose one

The inverse of an invertible lower triangular matrix is

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       ► lower triangular matrix                               

       ► upper triangular matrix                                

       ► diagonal matrix

Question No: 9      ( Marks: 1 ) - Please choose one

If P is a parallelepiped in R³, then

                        {volume of T (P)} = |detA|. {volume of P}

       ► Where T is determined by a  matrix A

       ► Where T is determined by a matrix A

       ► Where T is determined by a matrix A                           

       ► Where T is determined by a matrix A

Question No: 10      ( Marks: 1 ) - Please choose one

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Let A be a  matrix of rank  then row space of A has dimension

       ►

       ►

       ►

       ►

Question No: 11      ( Marks: 1 ) - Please choose one

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The dimension of the vector space  is

       ► 4

       ► 3

       ► 5

       ► 1

Question No: 12      ( Marks: 1 ) - Please choose one

Let .For the weighted Euclidean inner product  

       ► 2

       ► -2

       ► 3

       ► -3

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Question No: 13      ( Marks: 1 ) - Please choose one

Let A be  matrix whose entries are real. If  is an eigenvalue of A with x a corresponding eigenvector in , then

       ►

       ►

       ►

       ►

Question No: 14      ( Marks: 1 ) - Please choose one

Suppose that  has eigenvalues 2 and 0.5 .Then origin is a

       ► Saddle point

       ► Repellor

       ► Attractor

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Question No: 15      ( Marks: 1 ) - Please choose one

Which one is the numerical method used for approximation of dominant eigenvalue of a matrix.

       ► Power method

       ►  Jacobi’s method

       ► Guass Seidal method

       ► Gram Schmidt process    

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Question No: 16      ( Marks: 1 ) - Please choose one

The matrix equation represents a system of linear equations commonly referred to as the

       ► normal equations for

       ► normal equations for                                                             

       ► normal equations for

       ► normal equations for

Question No: 17      ( Marks: 1 ) - Please choose one

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Let have eigenvalues 2, 5, 0,-7, and -2. Then the dominant eigenvalue for A is

       ►

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Question No: 18      ( Marks: 1 ) - Please choose one

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If W is a subspace of , then the transformation  that maps each vector x in into its orthogonal x in W is called the orthogonal projection of

       ► in

       ► in W

       ► in x

Question No: 19      ( Marks: 1 ) - Please choose one

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If  , and then row reduction of   to

Produces a matrix P that satisfies

       ► for all x in V  

       ► for all x in V

       ► for all x in V  

       ► for all x in V 

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Question No: 20      ( Marks: 1 ) - Please choose one

The Casorati matrix for the signals 1^k, (-2)^k and 3^k is

       ►

       ►

       ►

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Question No: 21      ( Marks: 2 )

Find the characteristic polynomial and all eigenvalues of the given matrix

                                     

Question No: 22      ( Marks: 2 )

Write the Fourier coefficients  to the function  on the interval .

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Question No: 23      ( Marks: 2 )

The matrix A is followed by a sequence produced by the power method. Use these data to estimate the largest eigenvalue of A.

Question No: 24      ( Marks: 3 )

If   then find an invertible matrix P such that  

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Question No: 25      ( Marks: 3 )

Check whether the matrix has orthonormal columns or not?

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Question No: 26      ( Marks: 3 )

If A is a  matrix, what is the smallest possible dimension of Null A?

Question No: 27      ( Marks: 5 )

Assume that the matrix A is row equivalent to B. Without calculations, list rank A and dim Nul A. Then find bases for Col A and Row A.

Question No: 28      ( Marks: 5 )

Find an invertible matrix P and a matrix C of the form  such that the given matrix A has the form A=PCP^-1.

        with eigenvalue  and eigenvector

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Question No: 29      ( Marks: 5 )

Let  .Compute and compare Do not use Pythagorean Theorem.

Question No: 30      ( Marks: 10 )

Find a least squares solution of the inconsistent system Ax =b where .

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Question No: 31      ( Marks: 10 )

Determine whether the signals 1^k,2^k,and (-2)^k are the solutions of the difference equation

y_k+3 - y_k+2 - 4y_k+1 + 4y_k=0.