Linear Algebra - Determinants and Matrices
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Questions and Answers

What is the determinant of matrix A?

  • 45
  • 15
  • 1
  • 30 (correct)

    • Applying the operation '+ 3·r1 → r2' decreases the determinant.

    False

    What variable is present in the (4, 4) position of matrix L?

    h

    The projection matrix P projects a vector b onto Col(A). In this context, the projected vector b is ______.

    <p>[1, 4]</p> Signup and view all the answers

    Match the matrix operations with their effects on the determinant:

    <p>Row swap = Multiplies the determinant by -1 Row multiplication by a scalar = Multiplies the determinant by the scalar Adding a multiple of one row to another = Does not change the determinant Row scaling = Shifts the determinant by a power of the scale</p> Signup and view all the answers

    After the operation '15·r1 → r4', what is the immediate effect on the determinant?

    <p>It triples the determinant.</p> Signup and view all the answers

    The matrix A has a full rank if it is a square matrix with a non-zero determinant.

    <p>True</p> Signup and view all the answers

    What is the effect of performing the operation 'r4 + 7·r2 → r4'?

    <p>It modifies the fourth row of the matrix.</p> Signup and view all the answers

    In matrix A, the entry at position (2, 2) is ______.

    <p>2</p> Signup and view all the answers

    What is the projected vector b onto Col(A)?

    <p>[1, 4]</p> Signup and view all the answers

    Which column of matrix Y lies in the column space of matrix A?

    <p>Third column</p> Signup and view all the answers

    The matrix A presented has no missing entries.

    <p>False</p> Signup and view all the answers

    What result do you derive when multiplying the projection matrix onto Col(A) by matrix Y?

    <p>The resulting matrix after the operations as indicated in the content.</p> Signup and view all the answers

    The last column of Y is represented by the vector ______.

    <p>[4, 1, 2, 3]</p> Signup and view all the answers

    Match the following symbols or terms with their meanings in the context of matrix operations:

    <p>A = A matrix with missing entries Col(A) = The column space of matrix A Y = A matrix used in conjunction with A Null(A) = The null space of matrix A</p> Signup and view all the answers

    What is the trace of the matrix L?

    <p>12</p> Signup and view all the answers

    The vector u0 is defined as u0 = (a + 1).

    <p>False</p> Signup and view all the answers

    What is the polynomial f(t) given in the problem?

    <p>f(t) = t^9 - 15t^8 + 81t^7 - 179t^6 + 75t^5 + 303t^4 - 397t^3 - 9t^2 + 240t - 100</p> Signup and view all the answers

    The matrix A is defined in terms of a variable a, which is known to satisfy a ≃= ______.

    <p>1</p> Signup and view all the answers

    Match the following entries to their corresponding positions in the adjoint matrix adj(L):

    <p>(1, 5) = Unknown (5, 1) = Missing value (3, 3) = Known (2, 4) = To be determined</p> Signup and view all the answers

    Which entry in the polynomial f(t) is NOT an integer root?

    <p>-10</p> Signup and view all the answers

    Calculate the partial derivative of u1 with respect to a.

    <p>∂u1/∂a</p> Signup and view all the answers

    What is the primary focus of the problem regarding matrix A?

    <p>All of the above</p> Signup and view all the answers

    The scalar 2 cannot be an eigenvalue of matrix A if its corresponding eigenvector is v.

    <p>False</p> Signup and view all the answers

    What is the relationship between an eigenvector of a matrix and the adjugate of that matrix?

    <p>If v is an eigenvector of A corresponding to eigenvalue λ, then v is also an eigenvector of adj(A) with eigenvalue λ * det(A)/λ.</p> Signup and view all the answers

    The determinant of matrix A is given by det(A) = _____.

    <p>a + bi</p> Signup and view all the answers

    Match the following representations with their corresponding terminology:

    <p>det(A) = Determinant of matrix A trace(A) = Sum of eigenvalues adj(A) = Adjugate of matrix A eigenvector = Non-zero vector such that Av = λv</p> Signup and view all the answers

    What form does the Taylor series expansion of the matrix exponential exp(Bt) take?

    <p>0 1 t</p> Signup and view all the answers

    Eigenvalues can have multiple geometric multiplicities.

    <p>True</p> Signup and view all the answers

    What does the notation ωA(t) represent?

    <p>The function associated with the matrix A, typically indicating a characteristic polynomial or a matrix representation.</p> Signup and view all the answers

    The eigenvalue of A with the largest geometric multiplicity is ε = _____.

    <p>value based on matrix characteristics</p> Signup and view all the answers

    What dimensions does the matrix A have?

    <p>9 x 7</p> Signup and view all the answers

    The equation Av = 5 · w implies that the vector v is not a zero vector.

    <p>False</p> Signup and view all the answers

    What does it mean for the vector v to be an eigenvector of the Gramian of A?

    <p>It means that when the Gramian of A is multiplied by v, the result is a scalar multiple of v.</p> Signup and view all the answers

    The equation A↭ w = _____ · v indicates a relationship between w and v.

    <p>5</p> Signup and view all the answers

    Match the following vectors with their corresponding equations:

    <p>v = Av = 9 · w w = A↭ w = 5 · v</p> Signup and view all the answers

    Which of the following statements is true regarding the matrices and vectors in the equations?

    <p>v has 7 elements.</p> Signup and view all the answers

    If Av = 9 · w holds true, it indicates that v is a vector with non-zero elements.

    <p>False</p> Signup and view all the answers

    Identify the scalar multiple relating sine to the eigenvalue associated with v.

    <p>9</p> Signup and view all the answers

    The matrix used to find the eigenvalues associated with v is called the _____ of A.

    <p>Gramian</p> Signup and view all the answers

    What would happen if v does not equal zero in the equations provided?

    <p>v would no longer be an eigenvector.</p> Signup and view all the answers

    State the corresponding eigenvalue associated with the eigenvector v in the provided equations.

    <p>9</p> Signup and view all the answers

    Study Notes

    Exam I Instructions (General)

    • Exam duration: 50 minutes
    • 100 points
    • Number of problems: Varies by exam date (8, 6, 7, 5, 6 problems)
    • Show all work for credit.
    • Unsupported answers will not receive credit.
    • Scratch work will not be graded unless specifically requested in the problem

    Exam I Instructions (Specific to each exam)

    • Specific problem counts for each exam listed

    Exam II Instructions (General)

    • Exam duration: 50 minutes
    • 100 points
    • Number of problems: Varies by exam date (5, 5, 6, 5, 8 problems)
    • Show all work for credit.
    • Unsupported answers will not receive credit.
    • Scratch work will not be graded unless specifically requested in the problem

    Exam II Instructions (Specific to each exam)

    • Specific problem counts for each exam listed

    Exam III Instructions (General)

    • Exam duration: 50 minutes
    • 100 points
    • Number of problems: Varies by exam date (4, 4, 4, 6, 4 problems)
    • Show all work for credit.
    • Unsupported answers will not receive credit.
    • Scratch work will not be graded unless specifically requested in the problem

    Exam III Instructions (Specific to each exam)

    • Specific problem counts for each exam listed

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    Duke University Math 218D-2 Matrices and Vectors Exam I PDF

    Description

    This quiz explores key concepts of linear algebra, specifically focusing on determinants, matrix operations, and projection matrices. Questions cover the effects of various operations on determinants, the properties of matrices, and relationships between column spaces. Test your understanding of how matrix calculus influences outcomes in linear algebra.

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