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 determinant of a triangular matrix?

  1. The product of the diagonal elements

  2. The sum of the diagonal elements

  3. The difference of the diagonal elements

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The determinant of a triangular matrix can be calculated by simply multiplying the diagonal elements.

Multiple choice

What is the determinant of a matrix with a zero row or column?

  1. 0

  2. 1

  3. -1

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The determinant of a matrix with a zero row or column is always 0.

Multiple choice

What is the determinant of the transpose of a matrix?

  1. The same as the determinant of the original matrix

  2. The negative of the determinant of the original matrix

  3. The reciprocal of the determinant of the original matrix

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The determinant of the transpose of a matrix is equal to the determinant of the original matrix.

Multiple choice

What is the determinant of a matrix that is the product of two other matrices?

  1. The product of the determinants of the two matrices

  2. The sum of the determinants of the two matrices

  3. The difference of the determinants of the two matrices

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The determinant of a matrix that is the product of two other matrices is equal to the product of the determinants of those two matrices.

Multiple choice

What is the determinant of a matrix that is similar to another matrix?

  1. The same as the determinant of the original matrix

  2. The negative of the determinant of the original matrix

  3. The reciprocal of the determinant of the original matrix

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The determinant of a matrix that is similar to another matrix is equal to the determinant of the original matrix.

Multiple choice

What is the determinant of a matrix that is obtained by adding a multiple of one row to another row?

  1. The same as the determinant of the original matrix

  2. The negative of the determinant of the original matrix

  3. The determinant of the original matrix multiplied by the scalar

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Adding a multiple of one row to another row does not change the determinant of the matrix.

Multiple choice

What is the determinant of a matrix that is obtained by interchanging two rows?

  1. The same as the determinant of the original matrix

  2. The negative of the determinant of the original matrix

  3. The determinant of the original matrix multiplied by -1

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Interchanging two rows of a matrix changes the sign of the determinant.

Multiple choice

What is the determinant of a matrix that is obtained by multiplying a row by a scalar?

  1. The same as the determinant of the original matrix

  2. The negative of the determinant of the original matrix

  3. The determinant of the original matrix multiplied by the scalar

  4. None of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Multiplying a row of a matrix by a scalar multiplies the determinant by the same scalar.

Multiple choice

What is the determinant of a matrix that is obtained by adding a multiple of one column to another column?

  1. The same as the determinant of the original matrix

  2. The negative of the determinant of the original matrix

  3. The determinant of the original matrix multiplied by the scalar

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Adding a multiple of one column to another column does not change the determinant of the matrix.

Multiple choice

What is the determinant of a matrix that is obtained by interchanging two columns?

  1. The same as the determinant of the original matrix

  2. The negative of the determinant of the original matrix

  3. The determinant of the original matrix multiplied by -1

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Interchanging two columns of a matrix changes the sign of the determinant.

Multiple choice

What is the determinant of a matrix that is obtained by multiplying a column by a scalar?

  1. The same as the determinant of the original matrix

  2. The negative of the determinant of the original matrix

  3. The determinant of the original matrix multiplied by the scalar

  4. None of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Multiplying a column of a matrix by a scalar multiplies the determinant by the same scalar.

Multiple choice

Let (A) be a (3 \times 3) matrix with eigenvalues (1, 2, 3). Find the determinant of (A^2).

  1. 1

  2. 6

  3. 12

  4. 18
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The determinant of a matrix is the product of its eigenvalues. Therefore, the determinant of (A) is (1 \cdot 2 \cdot 3 = 6). The determinant of (A^2) is the square of the determinant of (A), so the determinant of (A^2) is (6^2 = 18).

Multiple choice

Let (A) be a (3 \times 3) matrix with eigenvalues (1, 2, 3). Find the trace of (A^3).

  1. 9

  2. 18

  3. 27

  4. 36
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The trace of a matrix is the sum of its eigenvalues. Therefore, the trace of (A) is (1 + 2 + 3 = 6). The trace of (A^3) is the trace of (A \cdot A \cdot A), which is equal to the sum of the eigenvalues of (A \cdot A \cdot A). The eigenvalues of (A \cdot A \cdot A) are the cubes of the eigenvalues of (A), which are (1^3, 2^3, 3^3). Therefore, the trace of (A^3) is (1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36).

Multiple choice

Let (A) be a (3 \times 3) matrix with eigenvalues (1, 2, 3). Find the determinant of (A^4).

  1. 1

  2. 81

  3. 243

  4. 729
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The determinant of a matrix is the product of its eigenvalues. Therefore, the determinant of (A) is (1 \cdot 2 \cdot 3 = 6). The determinant of (A^4) is the square of the determinant of (A^2), which is equal to the square of the determinant of (A \cdot A). The eigenvalues of (A \cdot A) are the squares of the eigenvalues of (A), which are (1^2, 2^2, 3^2). Therefore, the determinant of (A^2) is (1^2 \cdot 2^2 \cdot 3^2 = 36). The determinant of (A^4) is (36^2 = 729).

Multiple choice

Which of the following matrices represents a rotation of 45 degrees about the origin in the $xy$-plane?

  1. $$\begin{bmatrix} \cos 45° & -\sin 45° \\ \sin 45° & \cos 45° \end{bmatrix}$$
  2. $$\begin{bmatrix} \cos 45° & \sin 45° \\ -\sin 45° & \cos 45° \end{bmatrix}$$
  3. $$\begin{bmatrix} \cos 45° & 0 \\ 0 & \cos 45° \end{bmatrix}$$
  4. $$\begin{bmatrix} \cos 45° & 0 \\ 0 & -\cos 45° \end{bmatrix}$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The matrix $$\begin{bmatrix} \cos 45° & -\sin 45° \ \sin 45° & \cos 45° \end{bmatrix}$$ represents a rotation of 45 degrees about the origin in the $xy$-plane because it satisfies the following equation: $$\begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} \cos 45° & -\sin 45° \ \sin 45° & \cos 45° \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}$$, where $$(x', y')$$ are the coordinates of a point after the rotation and $$(x, y)$$ are the coordinates of the point before the rotation.