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Matrices

Description: Matrices Quiz: Test Your Understanding of Matrix Operations and Properties
Number of Questions: 14
Created by:
Tags: matrices matrix operations matrix properties linear algebra
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Given the matrices (A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}) and (B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}), find the product (AB).

  1. \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}

  2. \begin{bmatrix} 11 & 14 \ 25 & 30 \end{bmatrix}

  3. \begin{bmatrix} 17 & 20 \ 39 & 46 \end{bmatrix}

  4. \begin{bmatrix} 23 & 26 \ 51 & 58 \end{bmatrix}

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Correct Option: A
Explanation:

To find the product (AB), multiply the elements of the rows of (A) by the elements of the columns of (B) and add the products.

What is the determinant of the matrix (C = \begin{bmatrix} -2 & 3 \ 5 & -1 \end{bmatrix})?

  1. 1

  2. -1

  3. 5

  4. -5

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Correct Option: A
Explanation:

The determinant of a 2x2 matrix is calculated using the formula (ad - bc), where (a, b, c, d) are the elements of the matrix. In this case, the determinant of (C) is ((-2)(-1) - (3)(5) = 2 - 15 = -13).

Which of the following matrices is the identity matrix of order 3?

  1. \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}

  2. \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}

  3. \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}

  4. \begin{bmatrix} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{bmatrix}

Show answer
Correct Option: A
Explanation:

The identity matrix of order 3 is a square matrix with 1s on the diagonal and 0s everywhere else.

If (A) is a square matrix and (A^2 = 0), then which of the following statements is true?

  1. All eigenvalues of (A) are zero.

  2. All eigenvalues of (A) are non-zero.

  3. The determinant of (A) is zero.

  4. The trace of (A) is zero.

Show answer
Correct Option: C
Explanation:

If (A^2 = 0), then the determinant of (A) must be zero, as the determinant of a matrix is equal to the product of its eigenvalues.

What is the rank of the matrix (D = \begin{bmatrix} 1 & 2 & 3 \ 2 & 4 & 6 \ 3 & 6 & 9 \end{bmatrix})?

  1. 1

  2. 2

  3. 3

  4. 4

Show answer
Correct Option: A
Explanation:

The rank of a matrix is the maximum number of linearly independent rows or columns. In this case, the rows of (D) are linearly dependent, so the rank of (D) is 1.

Which of the following matrices is a symmetric matrix?

  1. \begin{bmatrix} 1 & 2 \ 2 & 3 \end{bmatrix}

  2. \begin{bmatrix} 1 & 2 \ -2 & 1 \end{bmatrix}

  3. \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}

  4. \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}

Show answer
Correct Option: C
Explanation:

A symmetric matrix is a square matrix that is equal to its transpose. In this case, only (\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}) is symmetric.

What is the trace of the matrix (E = \begin{bmatrix} 5 & 2 \ 3 & -1 \end{bmatrix})?

  1. 4

  2. 2

  3. 0

  4. -2

Show answer
Correct Option: A
Explanation:

The trace of a matrix is the sum of its diagonal elements. In this case, the trace of (E) is (5 + (-1) = 4).

If (F) is a square matrix and (F^T = -F), then which of the following statements is true?

  1. All eigenvalues of (F) are zero.

  2. All eigenvalues of (F) are non-zero.

  3. The determinant of (F) is zero.

  4. The trace of (F) is zero.

Show answer
Correct Option: C
Explanation:

If (F^T = -F), then the determinant of (F) must be zero, as the determinant of a matrix is equal to the product of its eigenvalues.

Which of the following matrices is an orthogonal matrix?

  1. \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}

  2. \begin{bmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} \ \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{bmatrix}

  3. \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}

  4. \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}

Show answer
Correct Option: A
Explanation:

An orthogonal matrix is a square matrix whose inverse is equal to its transpose. In this case, only (\begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}) is orthogonal.

What is the null space of the matrix (G = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix})?

  1. The set of all vectors (x) such that (Gx = 0).

  2. The set of all vectors (x) such that (xG = 0).

  3. The set of all vectors (x) such that (G^Tx = 0).

  4. The set of all vectors (x) such that (x^TG = 0).

Show answer
Correct Option: A
Explanation:

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector.

Which of the following matrices is a diagonalizable matrix?

  1. \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}

  2. \begin{bmatrix} 1 & 0 \ 0 & 2 \end{bmatrix}

  3. \begin{bmatrix} 1 & 2 \ 2 & 1 \end{bmatrix}

  4. \begin{bmatrix} 1 & 2 \ 3 & 6 \end{bmatrix}

Show answer
Correct Option: B
Explanation:

A diagonalizable matrix is a square matrix that can be expressed as a product of two matrices, one of which is a diagonal matrix. In this case, only (\begin{bmatrix} 1 & 0 \ 0 & 2 \end{bmatrix}) is diagonalizable.

What is the characteristic polynomial of the matrix (H = \begin{bmatrix} 2 & 3 \ -1 & 2 \end{bmatrix})?

  1. (x^2 - 4x + 7)

  2. (x^2 - 3x + 2)

  3. (x^2 + 4x + 7)

  4. (x^2 + 3x + 2)

Show answer
Correct Option: A
Explanation:

The characteristic polynomial of a matrix is a polynomial whose roots are the eigenvalues of the matrix. In this case, the characteristic polynomial of (H) is (x^2 - 4x + 7).

Which of the following matrices is a singular matrix?

  1. \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}

  2. \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}

  3. \begin{bmatrix} 1 & 2 \ 2 & 4 \end{bmatrix}

  4. \begin{bmatrix} 1 & 2 \ 3 & 6 \end{bmatrix}

Show answer
Correct Option: B
Explanation:

A singular matrix is a square matrix whose determinant is zero. In this case, only (\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}) is singular.

What is the cofactor of the element (a_{23}) of the matrix (J = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix})?

  1. -3

  2. 3

  3. -9

  4. 9

Show answer
Correct Option: C
Explanation:

The cofactor of an element (a_{ij}) of a matrix is the determinant of the submatrix obtained by deleting the (i)-th row and (j)-th column of the matrix, multiplied by ((-1)^{i+j}). In this case, the cofactor of (a_{23}) is ((-1)^{2+3}) times the determinant of the submatrix (\begin{bmatrix} 1 & 2 \ 7 & 8 \end{bmatrix}), which is (-9).

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