Mathematics

Linear Algebra

510 Questions

Linear algebra involves the study of matrices, vectors, and linear transformations. Common topics include finding the rank of a matrix, calculating eigenvalues and eigenvectors, and performing LU decomposition. These advanced mathematical concepts are frequently tested in engineering, statistics, and civil service examinations.

Matrix rank calculationLU decompositionEigenvalues and eigenvectorsMatrix invertibilityDeterminant properties

Linear Algebra Questions

Multiple choice

What is the characteristic polynomial of a matrix?

  1. A polynomial whose roots are the eigenvalues of the matrix

  2. A polynomial whose roots are the eigenvectors of the matrix

  3. A polynomial whose roots are the diagonal elements of the matrix

  4. A polynomial whose roots are the off-diagonal elements of the matrix
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Explanation

The characteristic polynomial of a matrix is a polynomial whose roots are the eigenvalues of the matrix.

Multiple choice

What is the Cayley-Hamilton theorem?

  1. A theorem that states that every square matrix satisfies its own characteristic polynomial

  2. A theorem that states that every square matrix satisfies its own minimal polynomial

  3. A theorem that states that every square matrix is similar to a diagonal matrix

  4. A theorem that states that every square matrix has at least one eigenvalue
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Explanation

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial.

Multiple choice

What is the Schur decomposition of a matrix?

  1. A decomposition of a matrix into a product of two unitary matrices

  2. A decomposition of a matrix into a product of two orthogonal matrices

  3. A decomposition of a matrix into a product of two diagonal matrices

  4. A decomposition of a matrix into a product of two triangular matrices
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Explanation

The Schur decomposition of a matrix is a decomposition of a matrix into a product of two unitary matrices.

Multiple choice

What is the singular value decomposition of a matrix?

  1. A decomposition of a matrix into a product of three matrices

  2. A decomposition of a matrix into a product of four matrices

  3. A decomposition of a matrix into a product of five matrices

  4. A decomposition of a matrix into a product of six matrices
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Explanation

The singular value decomposition of a matrix is a decomposition of a matrix into a product of three matrices.

Multiple choice

What is the QR decomposition of a matrix?

  1. A decomposition of a matrix into a product of two matrices

  2. A decomposition of a matrix into a product of three matrices

  3. A decomposition of a matrix into a product of four matrices

  4. A decomposition of a matrix into a product of five matrices
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A Correct answer
Explanation

The QR decomposition of a matrix is a decomposition of a matrix into a product of two matrices.

Multiple choice

What is the LU decomposition of a matrix?

  1. A decomposition of a matrix into a product of two matrices

  2. A decomposition of a matrix into a product of three matrices

  3. A decomposition of a matrix into a product of four matrices

  4. A decomposition of a matrix into a product of five matrices
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Explanation

The LU decomposition of a matrix is a decomposition of a matrix into a product of two matrices.

Multiple choice

What is the Cholesky decomposition of a matrix?

  1. A decomposition of a positive-definite matrix into a product of two triangular matrices

  2. A decomposition of a positive-definite matrix into a product of three triangular matrices

  3. A decomposition of a positive-definite matrix into a product of four triangular matrices

  4. A decomposition of a positive-definite matrix into a product of five triangular matrices
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Explanation

The Cholesky decomposition of a matrix is a decomposition of a positive-definite matrix into a product of two triangular matrices.

Multiple choice

What is the eigenvalue-eigenvector method for solving a system of linear differential equations?

  1. A method for solving a system of linear differential equations by finding the eigenvalues and eigenvectors of the coefficient matrix

  2. A method for solving a system of linear differential equations by finding the characteristic polynomial of the coefficient matrix

  3. A method for solving a system of linear differential equations by finding the minimal polynomial of the coefficient matrix

  4. A method for solving a system of linear differential equations by finding the Cayley-Hamilton theorem of the coefficient matrix
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Explanation

The eigenvalue-eigenvector method for solving a system of linear differential equations is a method for solving a system of linear differential equations by finding the eigenvalues and eigenvectors of the coefficient matrix.

Multiple choice

What is the power method for finding the largest eigenvalue and corresponding eigenvector of a matrix?

  1. A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a random vector

  2. A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is orthogonal to the previous vector

  3. A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is parallel to the previous vector

  4. A method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a vector that is equal to the previous vector
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Explanation

The power method for finding the largest eigenvalue and corresponding eigenvector of a matrix is a method for finding the largest eigenvalue and corresponding eigenvector of a matrix by repeatedly multiplying the matrix by a random vector.

Multiple choice

What is an eigenvalue of a square matrix?

  1. A scalar value associated with a corresponding eigenvector

  2. A vector that is parallel to the column space of the matrix

  3. The determinant of the matrix

  4. The trace of the matrix
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Explanation

An eigenvalue of a square matrix is a scalar value that, when substituted for the variable in the characteristic equation of the matrix, results in a nontrivial solution.

Multiple choice

What is an eigenvector of a square matrix?

  1. A nonzero vector that, when multiplied by the matrix, is scaled by the corresponding eigenvalue

  2. A vector that is orthogonal to the row space of the matrix

  3. The vector that corresponds to the largest eigenvalue of the matrix

  4. The vector that corresponds to the smallest eigenvalue of the matrix
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Explanation

An eigenvector of a square matrix is a nonzero vector that, when multiplied by the matrix, is scaled by the corresponding eigenvalue.

Multiple choice

What is the characteristic equation of a square matrix?

  1. An equation that is obtained by subtracting the identity matrix from the given matrix

  2. An equation that is obtained by adding the identity matrix to the given matrix

  3. An equation that is obtained by multiplying the given matrix by its transpose

  4. An equation that is obtained by subtracting the transpose of the given matrix from the identity matrix
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Explanation

The characteristic equation of a square matrix is an equation that is obtained by subtracting the identity matrix from the given matrix and setting the determinant of the resulting matrix equal to zero.

Multiple choice

What is the relationship between the eigenvalues and eigenvectors of a square matrix?

  1. Eigenvalues are the roots of the characteristic equation, and eigenvectors are the corresponding solutions to the homogeneous system of equations

  2. Eigenvalues are the roots of the characteristic equation, and eigenvectors are the corresponding solutions to the nonhomogeneous system of equations

  3. Eigenvalues are the solutions to the characteristic equation, and eigenvectors are the corresponding roots of the homogeneous system of equations

  4. Eigenvalues are the solutions to the characteristic equation, and eigenvectors are the corresponding roots of the nonhomogeneous system of equations
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Explanation

The eigenvalues of a square matrix are the roots of the characteristic equation, and the eigenvectors are the corresponding solutions to the homogeneous system of equations obtained by subtracting the eigenvalue from each diagonal entry of the matrix.

Multiple choice

What is the algebraic interpretation of an eigenvalue?

  1. It is the value that the matrix is multiplied by to obtain the identity matrix

  2. It is the value that the matrix is added to to obtain the identity matrix

  3. It is the value that the matrix is subtracted from to obtain the identity matrix

  4. It is the value that the matrix is divided by to obtain the identity matrix
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C Correct answer
Explanation

The algebraic interpretation of an eigenvalue is that it is the value that the matrix is subtracted from to obtain the identity matrix.

Multiple choice

What is the relationship between the eigenvalues and eigenvectors of a symmetric matrix?

  1. Eigenvalues are real and eigenvectors are orthogonal

  2. Eigenvalues are complex and eigenvectors are orthogonal

  3. Eigenvalues are real and eigenvectors are not orthogonal

  4. Eigenvalues are complex and eigenvectors are not orthogonal
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Explanation

For a symmetric matrix, the eigenvalues are real and the eigenvectors are orthogonal.