1.5.1 - Introduction and Types of Matrices.

1: What is a row matrix?

A) A matrix with only one column
B) A matrix with only one row
C) A matrix with equal number of rows and columns
D) A matrix where all elements are zero

Correct Answer: B) A matrix with only one row

2: What is a column matrix?

A) A matrix with only one column
B) A matrix with only one row
C) A matrix with equal number of rows and columns
D) A matrix where all elements are zero

Correct Answer: A) A matrix with only one column

3: What defines a square matrix?

A) A matrix where the number of rows is not equal to the number of columns
B) A matrix with equal number of rows and columns
C) A matrix with only one row
D) A matrix with only one column

Correct Answer: B) A matrix with equal number of rows and columns

4: What is a diagonal matrix?

A) A matrix where all elements are zero except those on the diagonal
B) A matrix where all elements are non-zero
C) A matrix where all diagonal elements are equal
D) A matrix with only one row

Correct Answer: A) A matrix where all elements are zero except those on the diagonal

5: What is a scalar matrix?

A) A diagonal matrix with all diagonal elements equal
B) A matrix where all elements are zero
C) A matrix with different elements on the diagonal
D) A matrix with non-zero elements below the diagonal

Correct Answer: A) A diagonal matrix with all diagonal elements equal

6: What defines a symmetric matrix?

A) A matrix where \(a_{ij} = -a_{ji}\)
B) A matrix where \(a_{ij} = a_{ji}\)
C) A matrix where all elements are zero
D) A matrix with only one column

Correct Answer: B) A matrix where \(a_{ij} = a_{ji}\)

7: What is a skew-symmetric matrix?

A) A matrix where \(a_{ij} = a_{ji}\)
B) A matrix where \(a_{ij} = -a_{ji}\)
C) A matrix where all elements are zero
D) A matrix where diagonal elements are zero

Correct Answer: B) A matrix where \(a_{ij} = -a_{ji}\)

8: What is an identity matrix?

A) A matrix with all diagonal elements equal to zero
B) A diagonal matrix with all diagonal elements equal to one
C) A matrix with all non-diagonal elements equal to one
D) A matrix with all elements zero except those below the diagonal

Correct Answer: B) A diagonal matrix with all diagonal elements equal to one

9: What defines an upper triangular matrix?

A) A matrix with all elements below the principal diagonal equal to zero
B) A matrix with all elements above the principal diagonal equal to zero
C) A matrix where all diagonal elements are zero
D) A matrix where all elements are non-zero

Correct Answer: A) A matrix with all elements below the principal diagonal equal to zero

10: What defines a lower triangular matrix?

A) A matrix with all elements below the principal diagonal equal to zero
B) A matrix with all elements above the principal diagonal equal to zero
C) A matrix where all diagonal elements are zero
D) A matrix with all elements non-zero

Correct Answer: B) A matrix with all elements above the principal diagonal equal to zero

11: What is a singular matrix?

A) A matrix where the determinant is zero
B) A matrix where the determinant is non-zero
C) A matrix with all elements zero
D) A matrix where all elements are equal

Correct Answer: A) A matrix where the determinant is zero

12: What is a non-singular matrix?

A) A matrix where the determinant is zero
B) A matrix where the determinant is non-zero
C) A matrix with all elements zero
D) A matrix where all diagonal elements are zero

Correct Answer: B) A matrix where the determinant is non-zero

13: When are two matrices considered equal?

A) When they have the same number of rows and columns
B) When their corresponding elements are equal
C) When their determinants are equal
D) When they are both diagonal matrices

Correct Answer: B) When their corresponding elements are equal

1.5.2 - Addition, Substraction and Scaller Multiplication of Matrices.

14: What is the result of adding two matrices A and B?

A) A matrix where each element is the product of the corresponding elements of A and B
B) A matrix where each element is the difference of the corresponding elements of A and B
C) A matrix where each element is the sum of the corresponding elements of A and B
D) A matrix where each element is the average of the corresponding elements of A and B

Correct Answer: C) A matrix where each element is the sum of the corresponding elements of A and B

15: What is the result of subtracting matrix B from matrix A?

A) A matrix where each element is the sum of the corresponding elements of A and B
B) A matrix where each element is the product of the corresponding elements of A and B
C) A matrix where each element is the difference of the corresponding elements of A and B
D) A matrix where each element is the quotient of the corresponding elements of A and B

Correct Answer: C) A matrix where each element is the difference of the corresponding elements of A and B

16: What is a negative matrix?

A) A matrix where all elements are positive
B) A matrix where all elements are zero
C) A matrix obtained by multiplying each element of the original matrix by -1
D) A matrix where each element is the reciprocal of the corresponding element of the original matrix

Correct Answer: C) A matrix obtained by multiplying each element of the original matrix by -1

17: Which law states that \(A + B = B + A\) for matrices?

A) Associative Law
B) Commutative Law
C) Additive Identity
D) Additive Inverse

Correct Answer: B) Commutative Law

18: Which law states that \(A + (B + C) = (A + B) + C\) for matrices?

A) Commutative Law
B) Associative Law
C) Additive Identity
D) Additive Inverse

Correct Answer: B) Associative Law

19: What is the additive identity matrix?

A) A matrix where all elements are 1
B) A matrix where all elements are -1
C) A matrix where all elements are 0
D) A matrix where all elements are equal

Correct Answer: C) A matrix where all elements are 0

20: What is the additive inverse of matrix A?

A) A matrix where each element is the reciprocal of the corresponding element of A
B) A matrix where each element is the negative of the corresponding element of A
C) A matrix where each element is the sum of the corresponding elements of A and B
D) A matrix where each element is the difference of the corresponding elements of A and B

Correct Answer: B) A matrix where each element is the negative of the corresponding element of A

21: What does the cancellation law for matrices state?

A) If \(A + B = A + C\), then \(B = C\)
B) If \(A + B = C\), then \(A = C - B\)
C) If \(A - B = A - C\), then \(B = C\)
D) If \(A + B = B + C\), then \(A = C\)

Correct Answer: A) If \(A + B = A + C\), then \(B = C\)

22: If matrices A and B are added and C is subtracted, which property ensures that \(A + (B - C) = (A + B) - C\)?

A) Commutative Law
B) Associative Law
C) Additive Identity
D) Additive Inverse

Correct Answer: B) Associative Law

1.5.3 - Multiplication of Matrices.

23: When can two matrices A and B be multiplied?

A) When the number of columns in A is equal to the number of rows in B
B) When the number of rows in A is equal to the number of columns in B
C) When both matrices have the same number of rows and columns
D) When both matrices are square matrices

Correct Answer: A) When the number of columns in A is equal to the number of rows in B

24: Which property of matrix multiplication states that \((AB)C = A(BC)\)?

A) Distributive Law
B) Associative Law
C) Identity Law
D) Commutative Law

Correct Answer: B) Associative Law

25: Which property of matrix multiplication states that \(A(B + C) = AB + AC\)?

A) Distributive Law
B) Associative Law
C) Identity Law
D) Commutative Law

Correct Answer: A) Distributive Law

26: Which property of matrix multiplication states that \((A + B)C = AC + BC\)?

A) Distributive Law
B) Associative Law
C) Identity Law
D) Commutative Law

Correct Answer: A) Distributive Law

27: What does the Identity Law for matrices state?

A) \(IA = A\) and \(AI = A\), where I is the identity matrix
B) \(IA = AI = 0\), where I is the identity matrix
C) \(IA = 0\) and \(AI = A\), where I is the identity matrix
D) \(IA = A\) and \(AI = 0\), where I is the identity matrix

Correct Answer: A) \(IA = A\) and \(AI = A\), where I is the identity matrix

28: If matrices A and B are such that the number of columns in A is equal to the number of rows in B, which operation is valid?

A) Matrix Addition
B) Matrix Subtraction
C) Matrix Multiplication
D) Matrix Transposition

Correct Answer: C) Matrix Multiplication

29: Which property of matrix multiplication is used to solve \((AB)C = A(BC)\)?

A) Distributive Law
B) Associative Law
C) Identity Law
D) Commutative Law

Correct Answer: B) Associative Law

30: Which property of matrix multiplication allows us to write \(A(B + C) = AB + AC\)?

A) Associative Law
B) Distributive Law
C) Identity Law
D) Commutative Law

Correct Answer: B) Distributive Law

31: Which property of matrix multiplication allows us to write \((A + B)C = AC + BC\)?

A) Commutative Law
B) Associative Law
C) Distributive Law
D) Identity Law

Correct Answer: C) Distributive Law

32: If \(A\) is a 2x3 matrix and \(B\) is a 3x4 matrix, what is the dimension of the product \(AB\)?

A) 2x3
B) 2x4
C) 3x4
D) 2x2

Correct Answer: B) 2x4

1.5.4 - Transpose of Matrix.

33: What is the transpose of a matrix?

A) A matrix obtained by multiplying each element by -1
B) A matrix formed by swapping the rows and columns of the original matrix
C) A matrix where each element is the sum of the corresponding elements of two matrices
D) A matrix where each element is the product of the corresponding elements of two matrices

Correct Answer: B) A matrix formed by swapping the rows and columns of the original matrix

34: Which of the following represents the transpose of matrix A?

A) A²
B) A'
C) A^-1
D) A!

Correct Answer: B) A'

35: Which property of the transpose states that \((A')' = A\)?

A) Transpose of the sum of matrices
B) Transpose of a scalar multiple
C) Transpose of a product of matrices
D) Double transpose

Correct Answer: D) Double transpose

36: Which property of the transpose states that \((kA)' = kA'\), where k is a scalar?

A) Transpose of the sum of matrices
B) Transpose of a scalar multiple
C) Transpose of a product of matrices
D) Double transpose

Correct Answer: B) Transpose of a scalar multiple

37: Which property of the transpose states that \((A + B)' = A' + B'\)?

A) Transpose of the sum of matrices
B) Transpose of a scalar multiple
C) Transpose of a product of matrices
D) Double transpose

Correct Answer: A) Transpose of the sum of matrices

38: Which property of the transpose states that \((AB)' = B'A'\)?

A) Transpose of the sum of matrices
B) Transpose of a scalar multiple
C) Transpose of a product of matrices
D) Double transpose

Correct Answer: C) Transpose of a product of matrices

39: If matrix A is a 3x2 matrix, what will be the dimensions of \(A'\) (the transpose of A)?

A) 3x2
B) 2x3
C) 2x2
D) 3x3

Correct Answer: B) 2x3

40: Given two matrices A and B, which of the following is true regarding their transposes?

A) \((AB)' = A'B'\)
B) \((AB)' = B'A'\)
C) \((AB)' = A' + B'\)
D) \((AB)' = A + B

Correct Answer: B) \((AB)' = B'A'\)

41: If \(A'\) is the transpose of matrix A, what is the result of transposing \(A'\) again?

A) A
B) \(A' + A\)
C) 2A
D) \(AA'\)

Correct Answer: A) A

42: What happens when the transpose of a matrix is multiplied by a scalar \(k\)?

A) The transpose is unaffected by the scalar
B) The scalar is distributed across all elements of the transpose
C) The scalar is only applied to the original matrix before taking the transpose
D) The scalar becomes the identity matrix

Correct Answer: B) The scalar is distributed across all elements of the transpose

1.5.5 - Rank of Matrix.

43: What is the rank of a matrix?

A) The number of rows in the matrix
B) The number of columns in the matrix
C) The largest order of any non-zero minor in the matrix
D) The sum of all elements in the matrix

Correct Answer: C) The largest order of any non-zero minor in the matrix

44: The rank of a matrix A is denoted by which symbol?

A) λ
B) ρ
C) σ
D) δ

Correct Answer: B) ρ

45: What must be true about the minors of order r+1 for a matrix to have rank r?

A) Every minor of order r+1 is non-zero
B) Every minor of order r+1 is zero
C) There is at least one non-zero minor of order r+1
D) There is at least one zero minor of order r+1

Correct Answer: B) Every minor of order r+1 is zero

46: For a matrix A to have a rank r, what condition must a minor of order r satisfy?

A) It must be singular
B) It must be non-singular
C) It must have all zeros
D) It must be equal to the determinant of the matrix

Correct Answer: B) It must be non-singular

47: Which of the following statements is true regarding the rank of a matrix?

A) The rank is always equal to the number of rows
B) The rank is always equal to the number of columns
C) The rank is the minimum of the number of rows and columns
D) The rank is the maximum order of a non-zero minor

Correct Answer: D) The rank is the maximum order of a non-zero minor

48: If every square sub-matrix of order r+1 is singular, what does this imply about the rank r of the matrix?

A) The rank is greater than r+1
B) The rank is less than or equal to r
C) The rank is exactly r+1
D) The rank is zero

Correct Answer: B) The rank is less than or equal to r

49: What does it mean if a matrix has a rank of zero?

A) All elements in the matrix are zero
B) The matrix is non-singular
C) The matrix is an identity matrix
D) The matrix has no rows or columns

Correct Answer: A) All elements in the matrix are zero

50: For a matrix A with rank r, what can be said about its square sub-matrix of order r+1?

A) It is always non-singular
B) It is always singular
C) It may be either singular or non-singular
D) It must have at least one non-zero element

Correct Answer: B) It is always singular

51: Which of the following is a correct method to determine the rank of a matrix?

A) Counting the number of non-zero rows
B) Counting the number of non-zero columns
C) Finding the maximum order of a non-zero minor
D) Summing all elements in the matrix

Correct Answer: C) Finding the maximum order of a non-zero minor

52: Which of the following conditions does not directly relate to determining the rank of a matrix?

A) Every square sub-matrix of order r+1 is singular
B) There is at least one non-singular sub-matrix of order r
C) The matrix has a non-zero determinant
D) Every minor of order r+1 is zero

Correct Answer: C) The matrix has a non-zero determinant