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

Which of the following matrices is invertible?

  1. [[1, 2], [3, 6]]

  2. [[2, 1], [1, 2]]

  3. [[0, 1], [0, 0]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

An invertible matrix is a square matrix that has an inverse matrix. The inverse matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. The matrix [[2, 1], [1, 2]] is invertible because its determinant is non-zero.

Multiple choice

What is the null space of the matrix [[1, 2], [3, 4]]?

  1. The set of all vectors x such that Ax = 0

  2. The set of all vectors x such that Ax = 1

  3. The set of all vectors x such that Ax = 2

  4. The set of all vectors x such that Ax = 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The null space of a matrix A is the set of all vectors x such that Ax = 0. This is also known as the kernel of the linear transformation represented by A.

Multiple choice

Which of the following is an orthogonal matrix?

  1. [[1, 0], [0, 1]]

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

  3. [[0, 1], [1, 0]]

  4. [[1, 2], [2, 1]]
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

An orthogonal matrix is a square matrix whose inverse is equal to its transpose. This means that the rows and columns of an orthogonal matrix are orthogonal to each other. The matrix [[1, 0], [0, 1]] is orthogonal because its inverse is also [[1, 0], [0, 1]].

Multiple choice

What is the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

  1. 1

  2. 2

  3. 3

  4. 4
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. The matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] has rank 2 because its first two rows are linearly independent, but its third row is a linear combination of the first two rows.

Multiple choice

Which of the following is a singular matrix?

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

  2. [[2, 1], [1, 2]]

  3. [[0, 1], [0, 0]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A singular matrix is a square matrix whose determinant is zero. The matrix [[0, 1], [0, 0]] is singular because its determinant is 0.

Multiple choice

What is the trace of the matrix [[1, 2], [3, 4]]?

  1. 5

  2. 7

  3. 9

  4. 11
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The trace of a matrix is the sum of its diagonal elements. The trace of the matrix [[1, 2], [3, 4]] is 1 + 4 = 5.

Multiple choice

Which of the following matrices is symmetric?

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

  2. [[2, 1], [1, 2]]

  3. [[0, 1], [1, 0]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A symmetric matrix is a square matrix that is equal to its transpose. The matrix [[2, 1], [1, 2]] is symmetric because its transpose is also [[2, 1], [1, 2]].

Multiple choice

What is the product of the matrices [[1, 2], [3, 4]] and [[5, 6], [7, 8]]?

  1. [[19, 22], [43, 50]]

  2. [[17, 20], [39, 46]]

  3. [[21, 24], [47, 54]]

  4. [[15, 18], [35, 42]]
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The product of two matrices is calculated by multiplying the elements of the rows of the first matrix by the elements of the columns of the second matrix and adding the products. The product of the matrices [[1, 2], [3, 4]] and [[5, 6], [7, 8]] is [[19, 22], [43, 50]].

Multiple choice

Which of the following is an elementary matrix?

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

  2. [[2, 1], [1, 2]]

  3. [[0, 1], [1, 0]]

  4. [[1, 0], [0, 1]]
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

An elementary matrix is a square matrix that can be obtained from the identity matrix by performing a single elementary row operation. The matrix [[0, 1], [1, 0]] is an elementary matrix because it can be obtained from the identity matrix by swapping the first and second rows.

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]]
Reveal answer Fill a bubble to check yourself
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
Reveal answer Fill a bubble to check yourself
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
Reveal answer Fill a bubble to check yourself
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.