Mathematics

Linear Algebra

449 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 adjoint of the matrix [[1, 2], [3, 4]]?

  1. [[4, -2], [-3, 1]]

  2. [[4, -3], [-2, 1]]

  3. [[4, -1], [-3, 2]]

  4. [[4, -2], [-1, 3]]
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A Correct answer
Explanation

The adjoint of a matrix is the transpose of its cofactor matrix. The cofactor matrix of a matrix is obtained by replacing each element of the matrix with the determinant of the submatrix formed by deleting the row and column containing that element. The adjoint of the matrix [[1, 2], [3, 4]] is [[4, -2], [-3, 1]].

Multiple choice

What is the process of finding a matrix that is similar to a given matrix called?

  1. Diagonalization

  2. Triangularization

  3. Orthogonalization

  4. Jordanization
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A Correct answer
Explanation

Diagonalization is the process of finding a matrix that is similar to a given matrix, meaning that they have the same eigenvalues and eigenvectors.

Multiple choice

What is an eigenvalue of a matrix?

  1. A scalar that, when multiplied by the matrix, produces the matrix itself

  2. A scalar that, when multiplied by the matrix, produces the zero matrix

  3. A scalar that, when multiplied by the matrix, produces a diagonal matrix

  4. A scalar that, when multiplied by the matrix, produces an orthogonal matrix
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An eigenvalue of a matrix is a scalar that, when multiplied by the matrix, produces a diagonal matrix.

Multiple choice

What is an eigenvector of a matrix?

  1. A vector that, when multiplied by the matrix, produces the eigenvalue of the matrix

  2. A vector that, when multiplied by the matrix, produces the zero vector

  3. A vector that, when multiplied by the matrix, produces a diagonal matrix

  4. A vector that, when multiplied by the matrix, produces an orthogonal matrix
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A Correct answer
Explanation

An eigenvector of a matrix is a vector that, when multiplied by the matrix, produces the eigenvalue of the matrix.

Multiple choice

What is the diagonalization theorem?

  1. A theorem that states that every square matrix can be diagonalized

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

  3. A theorem that states that every square matrix has at least one eigenvector

  4. A theorem that states that every square matrix is similar to a diagonal matrix
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The diagonalization theorem states that every square matrix is similar to a diagonal matrix.

Multiple choice

What is the Jordan canonical form of a matrix?

  1. A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal

  2. A matrix that is similar to a given matrix and has all of its eigenvectors as its columns

  3. A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal and all of its eigenvectors as its columns

  4. A matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal and all of its eigenvectors as its rows
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Jordan canonical form of a matrix is a matrix that is similar to a given matrix and has all of its eigenvalues on the diagonal.

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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A Correct answer
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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A Correct answer
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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A Correct answer
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
Reveal answer Fill a bubble to check yourself
A Correct answer
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
Reveal answer Fill a bubble to check yourself
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
Reveal answer Fill a bubble to check yourself
A Correct answer
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
Reveal answer Fill a bubble to check yourself
A Correct answer
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
Reveal answer Fill a bubble to check yourself
A Correct answer
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
Reveal answer Fill a bubble to check yourself
A Correct answer
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.