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    Unit 1 Essay Questions

    Uploaded by

    Tharun konda
    The document is a question bank for the subject Linear Algebra and Calculus from Visvodaya Technical Academy, Kavali. It contains 12 problems related to topics in linear algebra including: - Finding the rank of matrices by reducing them to echelon form. - Determining if a system of equations is consistent and solving it. - Finding the eigenvectors and eigenvalues of matrices. - Verifying and applying Cayley-Hamilton theorem. - Diagonalizing matrices and finding powers of matrices. - Defining concepts in linear algebra like triangular matrices, echelon form, and homogeneous/non-homogeneous systems of equations.

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    Unit 1 Essay Questions

    Uploaded by

    Tharun konda
    749 views2 pages
    The document is a question bank for the subject Linear Algebra and Calculus from Visvodaya Technical Academy, Kavali. It contains 12 problems related to topics in linear algebra including: - Finding the rank of matrices by reducing them to echelon form. - Determining if a system of equations is consistent and solving it. - Finding the eigenvectors and eigenvalues of matrices. - Verifying and applying Cayley-Hamilton theorem. - Diagonalizing matrices and finding powers of matrices. - Defining concepts in linear algebra like triangular matrices, echelon form, and homogeneous/non-homogeneous systems of equations.

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    The document is a question bank for the subject Linear Algebra and Calculus from Visvodaya Technical Academy, Kavali. It contains 12 problems related to topics in linear algebra including: - Finding the rank of matrices by reducing them to echelon form. - Determining if a system of equations is consistent and solving it. - Finding the eigenvectors and eigenvalues of matrices. - Verifying and applying Cayley-Hamilton theorem. - Diagonalizing matrices and finding powers of matrices. - Defining concepts in linear algebra like triangular matrices, echelon form, and homogeneous/non-homogeneous systems of equations.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    749 views2 pages

    Unit 1 Essay Questions

    Uploaded by

    Tharun konda
    The document is a question bank for the subject Linear Algebra and Calculus from Visvodaya Technical Academy, Kavali. It contains 12 problems related to topics in linear algebra including: - Finding the rank of matrices by reducing them to echelon form. - Determining if a system of equations is consistent and solving it. - Finding the eigenvectors and eigenvalues of matrices. - Verifying and applying Cayley-Hamilton theorem. - Diagonalizing matrices and finding powers of matrices. - Defining concepts in linear algebra like triangular matrices, echelon form, and homogeneous/non-homogeneous systems of equations.

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
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    VISVODAYA TECHNICAL ACADEMY, KAVALI

    DEPARTMENT OF MATHEMATICS
    I B.Tech
    QUESTION BANK

    Academic Year: 2020– 2021 Name of the Subject: Linear Algebra and Calculus
    *****************************************************************************************
    *
    UNIT-1
    Descriptive Questions:
    1 2 3 0
    1. a) Find the Rank of the Matrix A=
    [ ]
    2 4 3 2
    3 2 1 3
    6 8 7 5
    By reducing it to Echelon form.

    b) Prove that the set of given equations are consistent and solve them3 x+ 3 y +2 z=1, x +2 y =4 ,
    10 y +3 z=−2 ,2 x−3 y−z =5
    1 2 3 4

    2. a) Reduce the matrix A=


    [ 2 1 4 3
    3 0 5 −10 ]
    to its canonical form and find its Rank.
    b) Determine whether the following equations will have a nontrivial solution, If so Solve them.
    4 x  2 y  z  3w  0 , 6 x  3 y  4 z  7 w  0 , 2 x  y  w  0

    3. Find the Eigen values & Eigen vectors of the matrix:


     8 6 2 
     6 7  4 
     
     2 4 3 

    4. State Cayley Hamilton theorem, and verify Cayley Hamilton theorem for the matrix

    2 1 2

    A=
    [ 5 3 3
    −1 0 −2 ] and hence find A-1 , A4.

    8 −8 −2
    5. Diagonalizable the matrix A= 4 −3 −2
    3 −4 1 [ ] and Find A4

    6. a) Find whether the following equations are consistent, if so solve them. x  y  2 z  4 ,


    2 x  y  3 z  9 , 3x  y  z  2
    b) Prove that the sum of the Eigen values of a matrix is the trace of the matrix and product of the Eigen
    values is Determinant of the matrix.
     1 2 1 4 
    2 4 3 4
     
    1 2 3 4
     
    7. Reduce the following matrix to normal form and hence find its Rank   1 2 6 7 .
    8. State Cayley Hamilton theorem, and verify Cayley-Hamilton theorem for the matrix

    1 1 3

    A=
    [ 1 3 −3
    −2 −4 −4 ] and hence find A-1 .
    9. a) If is an Eigen value of A and X be the corresponding Eigen vector, then show that  n is the Eigen
    n
    value of A and X be the corresponding Eigen vector
    | A|
    b) If  is an Eigen value of A then prove that is an Eigen value of adjA
    ❑
    10.Define: i) Triangular Matrix ii) Echelon Form of a matrix
    iii) Normal Form of a matrix iv) Eigen value and Eigen vector
    v)Non-Homogeneous & Homogeneous system of linear equations Procedure (rules for consistency)
    vi) Diagonalization of a matrix
    11.For what values of  , μ The system of equation x + y + z=6 , x+ 2 y +3 z=10 , x+ 2 y + z=μ have

    i)No solution ii) Unique solution iii) Infinite solution

    1 0 0

    [
    12.Diagonalize the matrix A= 0 3 −1
    0 −1 3 ]

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