## 1-10: Fundamentals of Matrix Theory

### This set of (1-10) Matrices MCQS With Answers covers the basics of matrix theory, including determinants of 2x2 matrices, non-singular matrices and their inverses, properties of matrix multiplication, eigenvalues, and matrix rank determination.

#### 1. What is the sum of the elements in the first row of a 3x3 matrix?

- A) 0

- B) 1

- C) The sum cannot be determined from the given information.

- D) Depends on the specific matrix.

Correct Answer: A) 0

Explanation: The sum of the elements in the first row of any 3x3 matrix would be the sum of the three elements in that row, which can vary, but the sum itself can be any value. Thus, the correct answer is A) 0.

2. Which type of matrix has all its diagonal elements equal to zero?

- A) Identity Matrix

- B) Diagonal Matrix

- C) Symmetric Matrix

- D) Zero Matrix

Correct Answer: D) Zero Matrix

Explanation: In a zero matrix, all elements are zero, including the diagonal elements. This distinguishes it from other matrix types.

3. If matrix A is of order 3x4 and matrix B is of order 4x2, what is the order of the matrix AB?

- A) 3x2

- B) 4x3

- C) 3x4

- D) 4x2

Correct Answer: A) 3x2

Explanation: The product of matrices A and B will have the number of rows from matrix A and the number of columns from matrix B, resulting in a 3x2 matrix.

4. Which matrix operation is not commutative?

- A) Matrix Addition

- B) Matrix Multiplication

- C) Matrix Transposition

- D) Matrix Inversion

Correct Answer: B) Matrix Multiplication

Explanation: Matrix multiplication is not commutative. In most cases, AB ≠ BA, where A and B are matrices.

5. The determinant of an upper triangular matrix is the product of:

- A) Diagonal elements

- B) Off-diagonal elements

- C) Sum of elements

- D) Trace of the matrix

Correct Answer: A) Diagonal elements

Explanation: In an upper triangular matrix, all elements below the diagonal are zero, so the determinant is the product of the diagonal elements.

6. A matrix with a single row is called a:

- A) Column Matrix

- B) Row Matrix

- C) Square Matrix

- D) Diagonal Matrix

Correct Answer: B) Row Matrix

Explanation: A matrix with only one row is referred to as a row matrix.

7. Which property does the determinant of a square matrix satisfy when multiplied by its transpose?

- A) Associative Property

- B) Commutative Property

- C) Distributive Property

- D) Inverse Property

Correct Answer: B) Commutative Property

Explanation: For any square matrix A, det(A) * det (A^T) = det(A^T) * det(A), demonstrating the commutative property.

8. Inverse of a matrix exists if and only if its determinant is:

- A) Zero

- B) Non-negative

- C) Non-zero

- D) Negative

Correct Answer: C) Non-zero

Explanation: The inverse of a matrix exists if and only if its determinant is non-zero. Otherwise, the matrix is singular (non-invertible).

9. The LU decomposition of a matrix is used to simplify the process of finding its:

- A) Eigenvalues

- B) Determinant

- C) Inverse

- D) Transpose

Correct Answer: C) Inverse

Explanation: LU decomposition simplifies finding the inverse of a matrix by decomposing it into lower and upper triangular matrices.

10. Which property holds for the product of a matrix and the identity matrix?

- A) It’s the zero matrix.

- B) It’s the matrix itself.

- C) It’s the negative of the matrix.

- D) It’s the square of the matrix.

Correct Answer: B) It’s the matrix itself.

Explanation: When a matrix is multiplied by the identity matrix, the result is the matrix itself.

## 11-20: Advanced Concepts in Matrix Theory

Matrices Multiple Choice Questions With Answers Explanation |

### This set of (11-20) Matrix MCQs With Answers delves into more advanced concepts, such as orthogonal matrices, stochastic matrices, eigenvalues of diagonal matrices, the relationship between matrix properties and their determinants, and the nature of matrix products.

11. What is the result of matrix multiplication AB if matrix A is of order 2x3 and matrix B is of order 3x4?

- A) 2x4

- B) 3x3

- C) 2x3

- D) 3x4

Correct Answer: A) 2x4

Explanation: The result of matrix multiplication will have the number of rows from matrix A and the number of columns from matrix B, resulting in a 2x4 matrix.

12. Which type of matrix has the property that its transpose is equal to itself?

- A) Symmetric Matrix

- B) Diagonal Matrix

- C) Identity Matrix

- D) Skew-Symmetric Matrix

Correct Answer: A) Symmetric Matrix

Explanation: A symmetric matrix is equal to its transpose, i.e., A = A^T.

13. The determinant of a 2x2 matrix [a b; c d] is given by:

- A) ad

- B) bc

- C) ad – bc

- D) a + d

Correct Answer: C) ad – bc

Explanation: The determinant of a 2x2 matrix [a b; c d] is calculated as ad – bc.

14. What is the main diagonal of a matrix?

- A) The first row

- B) The first column

- C) The diagonal from the top-left to the bottom-right

- D) The diagonal from the top-right to the bottom-left

Correct Answer: C) The diagonal from the top-left to the bottom-right

Explanation: The main diagonal of a matrix consists of the elements running from the top-left corner to the bottom-right corner.

15. Which matrix operation involves swapping rows or columns to simplify calculations?

- A) Matrix Scaling

- B) Matrix Inversion

- C) Matrix Transposition

- D) Matrix Row Echelon Form

Correct Answer: D) Matrix Row Echelon Form

Explanation: The process of converting a matrix into row echelon form involves swapping rows to simplify calculations.

16. If the determinant of a square matrix is zero, the matrix is:

- A) Diagonal

- B) Invertible

- C) Singular

- D) Symmetric

Correct Answer: C) Singular

Explanation: A matrix with a determinant of zero is singular, meaning it’s not invertible.

17. The inverse of a matrix A is denoted by:

- A) A^T

- B) A^(-1)

- C) A’

- D) A^*

Correct Answer: B) A^(-1)

Explanation: The inverse of a matrix A is denoted as A^(-1).

18. The sum of a matrix and its negative is:

- A) A zero matrix

- B) A diagonal matrix

- C) A symmetric matrix

- D) A singular matrix

Correct Answer: A) A zero matrix

Explanation: Adding a matrix to its negative results in a zero matrix.

19. In LU decomposition, the lower triangular matrix contains:

- A) Zeros below the diagonal

- B) Zeros above the diagonal

- C) Non-zero elements below the diagonal

- D) Non-zero elements above the diagonal

Correct Answer: A) Zeros below the diagonal

Explanation: In LU decomposition, the lower triangular matrix contains zeros above the diagonal and non-zero elements below the diagonal.

20. If a matrix is both symmetric and diagonal, what type of matrix is it?

- A) Identity Matrix

- B) Skew-Symmetric Matrix

- C) Diagonal Matrix

- D) Zero Matrix

Correct Answer: C) Diagonal Matrix

Explanation: A diagonal matrix can be symmetric if its determinant elements are zero.

## 21-30: Properties of Symmetric Matrices

### This set of (21-30) Symmetric Matrices MCQS With Answers Explanation explores the properties of symmetric matrices, such as invertibility of matrices with distinct eigenvalues, trace and determinant relationships, and insights into eigenvalues of triangular matrices.

21. Which matrix operation involves swapping two rows of a matrix?

- A) Matrix Transposition

- B) Matrix Addition

- C) Matrix Row Interchange

- D) Matrix Multiplication

Correct Answer: C) Matrix Row Interchange

Explanation: Matrix row interchange involves swapping the positions of two rows in a matrix.

22. The determinant of a 3x3 matrix can be found using the:

- A) Formula ad – bc

- B) Cofactor expansion along any row or column

- C) Formula a^2 + b^2 + c^2

- D) Diagonal elements

Correct Answer: B) Cofactor expansion along any row or column

Explanation: The determinant of a 3x3 matrix can be found by using the cofactor expansion along any row or column.

23. The product of a matrix and its inverse is equal to the:

- A) Zero matrix

- B) Identity matrix

- C) Transpose of the matrix

- D) Diagonal matrix

Correct Answer: B) Identity matrix

Explanation: The product of a matrix and its inverse is the identity matrix, which is denoted by I.

24. If a matrix is both symmetric and skew-symmetric, what can be said about its elements?

- A) All elements are zero

- B) All diagonal elements are zero

- C) All off-diagonal elements are zero

- D) The matrix cannot be both symmetric and skew-symmetric

Correct Answer: A) All elements are zero

Explanation: A matrix cannot be both symmetric (A = A^T) and skew-symmetric (A = -A^T) unless all its elements are zero.

25. The inverse of an orthogonal matrix is:

- A) Symmetric

- B) Diagonal

- C) Orthogonal

- D) Not defined

Correct Answer: C) Orthogonal

Explanation: An orthogonal matrix is one where its transpose is also its inverse, making the matrix orthogonal.

26. What is the trace of a matrix?

- A) The sum of diagonal elements

- B) The sum of off-diagonal elements

- C) The product of diagonal elements

- D) The product of off-diagonal elements

Correct Answer: A) The sum of diagonal elements

Explanation: The trace of a matrix is the sum of its diagonal elements.

27. If matrix A is of order 4x3 and matrix B is of order 3x5, what is the order of the matrix BA?

- A) 4x5

- B) 3x4

- C) 4x3

- D) 3x5

Correct Answer: A) 4x5

Explanation: The product of matrices BA will have the number of rows from matrix B and the number of columns from matrix A, resulting in a 4x5 matrix.

28. Which matrix type has the property that its transpose is the negative of itself?

- A) Symmetric Matrix

- B) Skew-Symmetric Matrix

- C) Identity Matrix

- D) Diagonal Matrix

Correct Answer: B) Skew-Symmetric Matrix

Explanation: A skew-symmetric matrix is one where its transpose is the negative of itself.

29. The determinant of a matrix changes sign if:

- A) Two rows are swapped

- B) Two columns are swapped

- C) Any row is multiplied by a constant

- D) Any column is multiplied by a constant

Correct Answer: A) Two rows are swapped

Explanation: Swapping two rows in a matrix changes the sign of its determinant.

30. Which matrix operation involves multiplying a row by a non-zero constant?

- A) Matrix Scaling

- B) Matrix Transposition

- C) Matrix Row Echelon Form

- D) Matrix Inversion

Correct Answer: A) Matrix Scaling

Explanation: Matrix scaling involves multiplying a row of a matrix by a non-zero constant to simplify calculations.

## 31-40: Types of Matrices and Matrix Operations

### This set of (31-40) Matrix MCQs focuses on different types of matrices, such as identity, nilpotent, and upper triangular matrices, and matrix operations, such as transposition, row scaling, and elementary row operations for matrix solutions.

31. The rank of a matrix is determined by counting:

- A) The number of rows

- B) The number of columns

- C) The number of non-zero rows

- D) The number of non-zero columns

Correct Answer: C) The number of non-zero rows

Explanation: The rank of a matrix is the number of linearly independent rows (or columns) in the matrix.

32. Which matrix type is always non-singular and has a determinant of 1?

- A) Symmetric Matrix

- B) Orthogonal Matrix

- C) Diagonal Matrix

- D) Skew-Symmetric Matrix

Correct Answer: B) Orthogonal Matrix

Explanation: An orthogonal matrix is always non-singular and has a determinant of 1 or -1.

33. Which operation involves interchanging two columns of a matrix?

- A) Matrix Row Echelon Form

- B) Matrix Column Interchange

- C) Matrix Scaling

- D) Matrix Addition

Correct Answer: B) Matrix Column Interchange

Explanation: Matrix column interchange involves swapping the positions of two columns in a matrix.

34. A matrix is called idempotent if:

- A) Its determinant is zero

- B) It is invertible

- C) It is equal to its inverse

- D) It remains unchanged when squared

Correct Answer: D) It remains unchanged when squared

Explanation: A matrix is idempotent if squaring it results in the same matrix.

35. Which matrix operation involves subtracting a multiple of one row from another row?

- A) Matrix Scaling

- B) Matrix Row Addition

- C) Matrix Transposition

- D) Matrix Inversion

Correct Answer: B) Matrix Row Addition

Explanation: Matrix row addition involves subtracting a multiple of one row from another row to simplify calculations.

36. The eigenvalues of a diagonal matrix are:

- A) Always distinct

- B) Always zero

- C) The diagonal elements themselves

- D) Reciprocal of the diagonal elements

Correct Answer: C) The diagonal elements themselves

Explanation: The eigenvalues of a diagonal matrix are simply its diagonal elements.

37. Which property does the matrix multiplication hold with respect to scalar multiplication?

- A) Distributive Property

- B) Commutative Property

- C) Associative Property

- D) Inverse Property

Correct Answer: A) Distributive Property

Explanation: Matrix multiplication distributes over scalar multiplication, similar to how normal multiplication does.

38. The determinant of a 1x1 matrix [a] is:

- A) a

- B) 0

- C) 1

- D) Depends on the value of ‘a’

Correct Answer: A) a

Explanation: The determinant of a 1x1 matrix [a] is simply the value ‘a’.

39. Which matrix operation is involved in solving systems of linear equations?

- A) Matrix Transposition

- B) Matrix Row Echelon Form

- C) Matrix Scaling

- D) Matrix Inversion

Correct Answer: B) Matrix Row Echelon Form

Explanation: Matrix row echelon form is used to solve systems of linear equations.

40. The determinant of a skew-symmetric matrix of odd order is always:

- A) Zero

- B) Non-zero

- C) Negative

- D) Positive

Correct Answer: A) Zero

Explanation: The determinant of a skew-symmetric matrix of odd order is always zero.

## 41-50: Matrix Properties and Operations

### This set of (41-50) Matrices MCQs With Correct Answers covers matrix properties, such as zero and identity properties, operations to create upper triangular matrices, uniqueness of inverse for orthogonal matrices, and matrix scaling and column interchange operations.

41. Which matrix property states that the product of a matrix and the zero matrix is the zero matrix?

- A) Inverse Property

- B) Distributive Property

- C) Identity Property

- D) Zero Property

Correct Answer: D) Zero Property

Explanation: The product of any matrix and the zero matrix results in the zero matrix.

42. Which operation is used to create an upper triangular matrix from a given square matrix?

- A) Matrix Transposition

- B) Matrix Scaling

- C) Matrix Row Echelon Form

- D) Matrix Inversion

Correct Answer: C) Matrix Row Echelon Form

Explanation: The process of converting a matrix into row echelon form can lead to an upper triangular matrix.

43. The inverse of a matrix is unique if and only if the matrix is:

- A) Symmetric

- B) Diagonal

- C) Orthogonal

- D) Singular

Correct Answer: C) Orthogonal

Explanation: The inverse of an orthogonal matrix is unique, ensuring that there is only one inverse.

44. Which matrix operation involves multiplying a column by a non-zero constant?

- A) Matrix Scaling

- B) Matrix Transposition

- C) Matrix Column Echelon Form

- D) Matrix Inversion

Correct Answer: A) Matrix Scaling

Explanation: Matrix scaling involves multiplying a column of a matrix by a non-zero constant.

45. A matrix is called a nilpotent matrix if:

- A) Its determinant is zero

- B) It is invertible

- C) It is equal to its transpose

- D) It becomes the zero matrix when raised to a certain power

Correct Answer: D) It becomes the zero matrix when raised to a certain power

Explanation: A nilpotent matrix becomes the zero matrix when raised to a certain power.

46. Which matrix type is always symmetric and has all its diagonal elements equal to one?

- A) Identity Matrix

- B) Orthogonal Matrix

- C) Diagonal Matrix

- D) Zero Matrix

Correct Answer: A) Identity Matrix

Explanation: The identity matrix is always symmetric and has all its diagonal elements equal to one.

47. The matrix [1 0; 0 0] is an example of a:

- A) Symmetric Matrix

- B) Skew-Symmetric Matrix

- C) Diagonal Matrix

- D) Singular Matrix

Correct Answer: C) Diagonal Matrix

Explanation: The matrix [1 0; 0 0] is a diagonal matrix with all non-diagonal elements equal to zero.

48. The inverse of a skew-symmetric matrix is always:

- A) Symmetric

- B) Diagonal

- C) Skew-Symmetric

- D) Not defined

Correct Answer: C) Skew-Symmetric

Explanation: The inverse of a skew-symmetric matrix is also a skew-symmetric matrix.

49. Which matrix operation involves interchanging two columns of a matrix?

- A) Matrix Scaling

- B) Matrix Column Addition

- C) Matrix Transposition

- D) Matrix Column Interchange

Correct Answer: D) Matrix Column Interchange

Explanation: Matrix column interchange involves swapping the positions of two columns in a matrix.

50. If the determinant of a matrix is negative, which operation can change the sign of the determinant?

- A) Matrix Scaling

- B) Matrix Transposition

- C) Matrix Inversion

- D) Matrix Row Echelon Form

Correct Answer: C) Matrix Inversion

Explanation: Taking the inverse of a matrix with a negative determinant changes the sign of the determinant.

## 51-60: Hadamard Matrices and Other Topics

### This set of (51-60) Hadamard Matrices MCQS covers Hadamard matrices, matrix operations, such as column addition and column interchange, and the properties of adjugate matrices, block diagonal matrices, and the Kronecker product.

51. The product of two orthogonal matrices is:

- A) Orthogonal

- B) Symmetric

- C) Diagonal

- D) Non-singular

Correct Answer: A) Orthogonal

Explanation: The product of two orthogonal matrices is itself orthogonal.

52. A matrix with all its elements equal to a constant is called a:

- A) Diagonal Matrix

- B) Identity Matrix

- C) Scalar Matrix

- D) Zero Matrix

Correct Answer: C) Scalar Matrix

Explanation: A scalar matrix is one where all diagonal elements are equal, and all off-diagonal elements are zero.

53. Which matrix property states that the order of multiplication matters?

- A) Distributive Property

- B) Associative Property

- C) Commutative Property

- D) Inverse Property

Correct Answer: B) Associative Property

Explanation: The associative property of matrix multiplication states that the order of multiplication matters.

54. A matrix with only one column is called a:

- A) Row Matrix

- B) Square Matrix

- C) Column Matrix

- D) Diagonal Matrix

Correct Answer: C) Column Matrix

Explanation: A matrix with only one column is referred to as a column matrix.

55. The determinant of a block diagonal matrix is the product of the determinants of its:

- A) Diagonal blocks

- B) Off-diagonal blocks

- C) Upper triangular blocks

- D) Lower triangular blocks

Correct Answer: A) Diagonal blocks

Explanation: The determinant of a block diagonal matrix is the product of the determinants of its diagonal blocks.

56. Which operation involves swapping two columns of a matrix?

- A) Matrix Column Interchange

- B) Matrix Scaling

- C) Matrix Transposition

- D) Matrix Column Echelon Form

Correct Answer: A) Matrix Column Interchange

Explanation: Matrix column interchange involves swapping the positions of two columns in a matrix.

57. The adjugate of a matrix is used to calculate its:

- A) Rank

- B) Trace

- C) Inverse

- D) Determinant

Correct Answer: C) Inverse

Explanation: The adjugate of a matrix is used in finding the inverse of a matrix using the formula A^(-1) = adj(A) / det(A).

58. A matrix is called a stochastic matrix if:

- A) All its elements are integers

- B) It has at least one zero row

- C) All its row sums are equal to one

- D) Its determinant is one

Correct Answer: C) All its row sums are equal to one

Explanation: A stochastic matrix is one where all its row sums are equal to one.

59. The eigenvalues of a triangular matrix are:

- A) Always distinct

- B) Always zero

- C) The diagonal elements themselves

- D) Reciprocal of the diagonal elements

Correct Answer: C) The diagonal elements themselves

Explanation: The eigenvalues of a triangular matrix are its diagonal elements.

60. Which matrix operation involves replacing a row by the sum of itself and a multiple of another row?

- A) Matrix Row Addition

- B) Matrix Column Interchange

- C) Matrix Transposition

- D) Matrix Scaling

Correct Answer: A) Matrix Row Addition

Explanation: Matrix row addition involves replacing a row by the sum of itself and a multiple of another row to simplify calculations.

## 61-70: Matrix Transformations and Norms

### This set of (61-70) Matrix Transformation MCQs explores row interchange and transposition operations, lower triangular matrix creation, the Frobenius norm as a measure of magnitude, and applications of matrix row echelon form transformations.

61. Which matrix operation involves replacing a column by the sum of itself and a multiple of another column?

- A) Matrix Column Addition

- B) Matrix Column Interchange

- C) Matrix Transposition

- D) Matrix Scaling

Correct Answer: A) Matrix Column Addition

Explanation: Matrix column addition involves replacing a column by the sum of itself and a multiple of another column to simplify calculations.

62. A matrix with only one non-zero element is called a:

- A) Diagonal Matrix

- B) Identity Matrix

- C) Sparse Matrix

- D) Zero Matrix

Correct Answer: C) Sparse Matrix

Explanation: A sparse matrix is one that has very few non-zero elements.

63. The inverse of a lower triangular matrix is:

- A) Always lower triangular

- B) Upper triangular

- C) Diagonal

- D) Not defined

Correct Answer: A) Always lower triangular

Explanation: The inverse of a lower triangular matrix is always lower triangular.

64. Which matrix operation involves replacing a row by the sum of itself and a multiple of another row, and applying it to another row?

- A) Matrix Row Scaling

- B) Matrix Row Echelon Form

- C) Matrix Row Addition

- D) Matrix Transposition

Correct Answer: C) Matrix Row Addition

Explanation: Matrix row addition involves replacing a row by the sum of itself and a multiple of another row, and applying it to another row.

65. The Frobenius norm of a matrix is a measure of its:

- A) Determinant

- B) Rank

- C) Magnitude

- D) Trace

Correct Answer: C) Magnitude

Explanation: The Frobenius norm of a matrix is a measure of its magnitude or size.

66. Which matrix operation involves applying a sequence of elementary row operations to transform a matrix?

- A) Matrix Row Addition

- B) Matrix Row Echelon Form

- C) Matrix Scaling

- D) Matrix Transposition

Correct Answer: B) Matrix Row Echelon Form

Explanation: The process of applying elementary row operations to a matrix results in its row echelon form.

67. The transpose of the inverse of a matrix is equal to the:

- A) Transpose of the matrix

- B) Inverse of the transpose of the matrix

- C) Identity matrix

- D) Zero matrix

Correct Answer: B) Inverse of the transpose of the matrix

Explanation: (A^(-1))^T = (A^T)^(-1) holds true for any invertible matrix A.

68. The eigenvalues of an upper triangular matrix are:

- A) Always distinct

- B) Always zero

- C) The diagonal elements themselves

- D) Reciprocal of the diagonal elements

Correct Answer: C) The diagonal elements themselves

Explanation: The eigenvalues of an upper triangular matrix are its diagonal elements.

69. Which matrix property states that the product of a matrix and the identity matrix leaves the matrix unchanged?

- A) Inverse Property

- B) Distributive Property

- C) Identity Property

- D) Zero Property

Correct Answer: C) Identity Property

Explanation: The product of any matrix and the identity matrix results in the original matrix.

70. The determinant of a 2x2 matrix [a b; c d] is the same as the determinant of its transpose:

- A) Always

- B) Sometimes

- C) Never

- D) Depends on the values of a, b, c, and d

Correct Answer: A) Always

Explanation: The determinant of a 2x2 matrix and its transpose are always the same.

## 71-80: Skew-Symmetric Matrices and Singular Matrices

### This set of (71-80) MCQs addresses skew-symmetric matrices, orthogonal matrix characteristics, singular matrices, symmetric matrix eigenvalues, and the inverse of diagonal matrices.

71. Which matrix operation involves multiplying all elements of a matrix by a non-zero constant?

- A) Matrix Scaling

- B) Matrix Transposition

- C) Matrix Row Echelon Form

- D) Matrix Inversion

Correct Answer: A) Matrix Scaling

Explanation: Matrix scaling involves multiplying all elements of a matrix by a non-zero constant.

72. Which matrix operation is equivalent to swapping rows and columns of a matrix?

- A) Matrix Transposition

- B) Matrix Row Interchange

- C) Matrix Column Interchange

- D) Matrix Scaling

Correct Answer: A) Matrix Transposition

Explanation: Transposing a matrix involves swapping its rows and columns.

73. A matrix is called a Hadamard matrix if:

- A) Its determinant is one

- B) It is orthogonal

- C) All its elements are non-negative

- D) Its rows are orthogonal and have equal magnitudes

Correct Answer: D) Its rows are orthogonal and have equal magnitudes

Explanation: A Hadamard matrix is one where its rows are orthogonal and have equal magnitudes (lengths).

74. The determinant of a block matrix is the product of the determinants of its:

- A) Blocks along the diagonal

- B) Blocks along the anti-diagonal

- C) Off-diagonal blocks

- D) Upper triangular blocks

Correct Answer: A) Blocks along the diagonal

Explanation: The determinant of a block matrix is the product of the determinants of its diagonal blocks.

75. Which matrix property states that the order of addition matters?

- A) Distributive Property

- B) Associative Property

- C) Commutative Property

- D) Inverse Property

Correct Answer: C) Commutative Property

Explanation: The commutative property of matrix addition states that the order of addition matters.

76. The eigenvalues of a matrix are found by solving the equation:

- A) A * x = - B) A * x = x

- C) det(A) = 0

- D) det(A) = 1

Correct Answer: B) A * x = x

Explanation: The eigenvalues of a matrix A are the values λ for which A * x = λ * x, where x is a non-zero vector.

77. The diagonal elements of a diagonal matrix are the:

- A) Eigenvalues

- B) Row sums

- C) Column sums

- D) Trace

Correct Answer: A) Eigenvalues

Explanation: The diagonal elements of a diagonal matrix are its eigenvalues.

78. Which matrix operation involves adding a constant multiple of one column to another column?

- A) Matrix Column Scaling

- B) Matrix Column Addition

- C) Matrix Transposition

- D) Matrix Inversion

Correct Answer: B) Matrix Column Addition

Explanation: Matrix column addition involves adding a constant multiple of one column to another column to simplify calculations.

79. Which matrix type is always invertible?

- A) Symmetric Matrix

- B) Diagonal Matrix

- C) Singular Matrix

- D) Orthogonal Matrix

Correct Answer: D) Orthogonal Matrix

Explanation: An orthogonal matrix is always invertible.

80. The inverse of a non-singular matrix is:

- A) Singular

- B) Skew-Symmetric

- C) Orthogonal

- D) Non-singular

Correct Answer: D) Non-singular

Explanation: The inverse of a non-singular matrix is itself non-singular.

## 80-90: Determinants and Matrix Equations

### In this set of 80-90 MCQS, you’ll learn about the determinant of 2x2 matrices and their transposes, skew-symmetric matrix inverses, the commutative property of matrix addition, and the characteristic equation’s role in finding eigenvalues.

81. The Kronecker product of two matrices results in a matrix that is:

- A) Symmetric

- B) Diagonal

- C) Orthogonal

- D) Block Matrix

Correct Answer: D) Block Matrix

Explanation: The Kronecker product of two matrices results in a block matrix.

82. Which matrix property states that the product of a matrix and its inverse is the identity matrix?

- A) Inverse Property

- B) Distributive Property

- C) Identity Property

- D) Zero Property

Correct Answer: A) Inverse Property

Explanation: The inverse property of matrix multiplication states that the product of a matrix and its inverse is the identity matrix.

83. The inverse of a skew-symmetric matrix of odd order is:

- A) Always skew-symmetric

- B) Skew-symmetric if the determinant is -1

- C) Skew-symmetric if the determinant is 1

- D) Not defined

Correct Answer: A) Always skew-symmetric

Explanation: The inverse of a skew-symmetric matrix of odd order is always skew-symmetric.

84. Which matrix operation involves replacing a column by a non-zero multiple of itself and adding it to another column?

- A) Matrix Scaling

- B) Matrix Column Addition

- C) Matrix Transposition

- D) Matrix Column Echelon Form

Correct Answer: B) Matrix Column Addition

Explanation: Matrix column addition involves replacing a column by a non-zero multiple of itself and adding it to another column.

85. A matrix is called orthogonal if its columns are:

- A) Linearly dependent

- B) Linearly independent

- C) Orthogonal

- D) Skew-Symmetric

Correct Answer: B) Linearly independent

Explanation: An orthogonal matrix has linearly independent columns.

86. The rank of a zero matrix is always:

- A) Zero

- B) One

- C) The number of rows

- D) The number of columns

Correct Answer: A) Zero

Explanation: The rank of a zero matrix is always zero.

87. The determinant of the identity matrix is:

- A) Zero

- B) One

- C) The number of rows

- D) The number of columns

Correct Answer: B) One

Explanation: The determinant of the identity matrix is always one.

88. The product of two diagonal matrices is a matrix with:

- A) Diagonal elements as the sum of diagonal elements of the two matrices

- B) Diagonal elements as the product of diagonal elements of the two matrices

- C) Diagonal elements as the difference of diagonal elements of the two matrices

- D) Diagonal elements as the average of diagonal elements of the two matrices

Correct Answer: B) Diagonal elements as the product of diagonal elements of the two matrices

Explanation: The product of two diagonal matrices results in a matrix with diagonal elements as the product of diagonal elements of the two matrices.

89. The inverse of a skew-symmetric matrix is:

- A) Symmetric

- B) Diagonal

- C) Skew-Symmetric

- D) Not defined

Correct Answer: C) Skew-Symmetric

Explanation: The inverse of a skew-symmetric matrix is also skew-symmetric.

90. Which matrix type is always orthogonal and has determinant 1?

- A) Symmetric Matrix

- B) Identity Matrix

- C) Diagonal Matrix

- D) Orthogonal Matrix

Correct Answer: D) Orthogonal Matrix

Explanation: An orthogonal matrix is always orthogonal and has a determinant of 1 or -1.

## 90-100: Applications of Matrix Theory

### This set of 90-100 MCQS covers matrix scaling, rank determination, orthogonal matrix products, stochastic matrices, diagonal matrix eigenvalues, and the relationship between diagonal matrix inverses and their elements.

91. Which matrix property states that the product of a matrix and the zero vector is the zero vector?

- A) Inverse Property

- B) Distributive Property

- C) Zero Property

- D) Identity Property

Correct Answer: C) Zero Property

Explanation: The product of any matrix and the zero vector results in the zero vector.

92. The characteristic equation of a matrix is used to find its:

- A) Rank

- B) Eigenvalues

- C) Determinant

- D) Inverse

Correct Answer: B) Eigenvalues

Explanation: The characteristic equation is used to find the eigenvalues of a matrix.

93. The matrix [0 1; 1 0] is an example of a:

- A) Symmetric Matrix

- B) Skew-Symmetric Matrix

- C) Diagonal Matrix

- D) Orthogonal Matrix

Correct Answer: D) Orthogonal Matrix

Explanation: The matrix [0 1; 1 0] is an orthogonal matrix since its transpose is its inverse.

94. Which matrix operation involves adding a constant multiple of one row to another row?

- A) Matrix Row Scaling

- B) Matrix Row Addition

- C) Matrix Transposition

- D) Matrix Inversion

Correct Answer: B) Matrix Row Addition

Explanation: Matrix row addition involves adding a constant multiple of one row to another row to simplify calculations.

95. Which operation is used to create a lower triangular matrix from a given square matrix?

- A) Matrix Transposition

- B) Matrix Scaling

- C) Matrix Column Echelon Form

- D) Matrix Row Echelon Form

Correct Answer: C) Matrix Column Echelon Form

Explanation: The process of converting a matrix into column echelon form can lead to a lower triangular matrix.

96. The eigenvalues of a matrix are also called its:

- A) Diagonal elements

- B) Determinants

- C) Characteristic roots

- D) Row sums

Correct Answer: C) Characteristic roots

Explanation: The eigenvalues of a matrix are also referred to as its characteristic roots.

97. Which matrix operation is equivalent to swapping rows and columns of a matrix and then taking the transpose?

- A) Matrix Transposition

- B) Matrix Row Interchange

- C) Matrix Column Interchange

- D) Matrix Scaling

Correct Answer: A) Matrix Transposition

Explanation: Transposing a matrix involves swapping its rows and columns, which is equivalent to interchanging rows and columns followed by taking the transpose.

98. The diagonal elements of a symmetric matrix are the:

- A) Eigenvalues

- B) Row sums

- C) Column sums

- D) Trace

Correct Answer: A) Eigenvalues

Explanation: The diagonal elements of a symmetric matrix are its eigenvalues.

99. Which matrix type is always singular?

- A) Symmetric Matrix

- B) Identity Matrix

- C) Diagonal Matrix

- D) Skew-Symmetric Matrix

Correct Answer: B) Identity Matrix

Explanation: The determinant of the identity matrix is always 1, making it non-singular.

100. The inverse of a diagonal matrix is obtained by:

- A) Taking the reciprocal of each diagonal element

- B) Swapping the diagonal elements

- C) Adding the diagonal elements

- D) Taking the negative of each diagonal element

Correct Answer: A) Taking the reciprocal of each diagonal element

Explanation: The inverse of a diagonal matrix is obtained by taking the reciprocal of each diagonal element.