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Unit 1: Matrix Algebra - Practice Quiz

MTH174 — Engineering Mathematics 60 Questions
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1 Which of the following is a valid elementary row operation on a matrix?

elementary operations and their use in getting the rank Easy
A. Adding a constant to all elements of a row
B. Squaring all elements of a row
C. Interchanging any two rows
D. Multiplying a row by zero

2 The rank of a matrix in row echelon form is equal to the number of:

elementary operations and their use in getting the rank Easy
A. Columns
B. Non-zero rows
C. Zero rows
D. Total rows

3 What is the rank of a null (zero) matrix of order ?

elementary operations and their use in getting the rank Easy
A.
B.
C.
D. Undefined

4 Do elementary operations change the rank of a matrix?

elementary operations and their use in getting the rank Easy
A. No, elementary operations do not alter the rank.
B. No, unless the operation involves multiplying by a negative number.
C. Yes, they always change the rank.
D. Yes, but only column operations change it.

5 A square matrix has an inverse if and only if its determinant is:

inverse of a matrix and solution of linear simultaneous equations Easy
A. Not equal to
B. Equal to
C. Negative
D. Equal to

6 What is the inverse of the identity matrix ?

inverse of a matrix and solution of linear simultaneous equations Easy
A. It does not exist
B. The null matrix
C.
D. The identity matrix

7 For two invertible matrices and , which of the following is true for the inverse of their product, ?

inverse of a matrix and solution of linear simultaneous equations Easy
A.
B.
C.
D.

8 If is an invertible matrix and is its inverse, what does the product yield?

inverse of a matrix and solution of linear simultaneous equations Easy
A. The identity matrix
B. A diagonal matrix with zeroes on the diagonal
C. The matrix
D. The null matrix

9 A system of linear equations is called a homogeneous system if:

inverse of a matrix and solution of linear simultaneous equations Easy
A.
B.
C.
D.

10 In a system of linear equations , the system is called inconsistent if:

inverse of a matrix and solution of linear simultaneous equations Easy
A. The matrix is singular
B. It has no solution
C. It has a unique solution
D. It has infinite solutions

11 A homogeneous system of linear equations always has at least one solution. What is this solution called?

inverse of a matrix and solution of linear simultaneous equations Easy
A. The singular solution
B. The trivial solution
C. The inconsistent solution
D. The dominant solution

12 For a system of equations in variables , if , the system has:

inverse of a matrix and solution of linear simultaneous equations Easy
A. Infinite solutions
B. No solution
C. A unique solution
D. Exactly two solutions

13 The characteristic equation of a square matrix is given by:

eigen-values and eigenvectors of a matrix Easy
A.
B.
C.
D.

14 The sum of the eigenvalues of a square matrix is always equal to:

eigen-values and eigenvectors of a matrix Easy
A. Zero
B. The trace of the matrix (sum of principal diagonal elements)
C. The determinant of the matrix
D. One

15 The product of the eigenvalues of a square matrix is equal to:

eigen-values and eigenvectors of a matrix Easy
A. The trace of matrix
B. The inverse of matrix
C. The rank of matrix
D. The determinant of matrix

16 For a lower triangular matrix, the eigenvalues are simply the elements on its:

eigen-values and eigenvectors of a matrix Easy
A. Last row
B. Secondary diagonal
C. First column
D. Principal diagonal

17 The Cayley-Hamilton theorem states that every square matrix satisfies its own:

Cayley-Hamilton theorem Easy
A. Determinant
B. Characteristic equation
C. Inverse
D. Transpose

18 Which of the following matrices does the Cayley-Hamilton theorem apply to?

Cayley-Hamilton theorem Easy
A. Any square matrix
B. Only singular matrices
C. Any rectangular matrix
D. Only matrices

19 If the characteristic equation of a matrix is , then according to the Cayley-Hamilton theorem:

Cayley-Hamilton theorem Easy
A.
B.
C.
D.

20 The Cayley-Hamilton theorem is commonly used in matrix algebra to efficiently find:

Cayley-Hamilton theorem Easy
A. The transpose of a matrix
B. The inverse and higher powers of a matrix
C. The trace of a matrix
D. The rank of a matrix

21 Which of the following statements is true regarding the rank of a matrix when subjected to elementary row operations?

elementary operations and their use in getting the rank Medium
A. Elementary row operations can decrease the rank of a matrix.
B. Elementary row operations do not alter the rank of a matrix.
C. Elementary row operations can increase the rank of a matrix.
D. The rank becomes zero after a finite number of elementary operations.

22 If a matrix has all its elements equal to 1, what is the rank of ?

elementary operations and their use in getting the rank Medium
A.
B.
C.
D.

23 For a square matrix of order , if the determinant , what can be said about its rank ?

elementary operations and their use in getting the rank Medium
A.
B.
C.
D.

24 If the rank of a matrix is , how many linearly independent rows does it have?

elementary operations and their use in getting the rank Medium
A.
B.
C.
D.

25 When finding the inverse of a matrix using Gauss-Jordan elimination, we apply elementary row operations to the augmented matrix until it becomes:

inverse of a matrix and solution of linear simultaneous equations Medium
A.
B.
C.
D.

26 If matrices and are invertible matrices of the same order, then is equal to:

inverse of a matrix and solution of linear simultaneous equations Medium
A.
B.
C.
D.

27 Consider a system of linear equations with variables. If the rank of the coefficient matrix is equal to the rank of the augmented matrix but less than , the system has:

inverse of a matrix and solution of linear simultaneous equations Medium
A. Only the trivial solution
B. A unique solution
C. No solution
D. Infinitely many solutions

28 For a homogeneous system of linear equations of order , the system has non-trivial solutions if and only if:

inverse of a matrix and solution of linear simultaneous equations Medium
A.
B. Rank of
C.
D.

29 If is an eigenvalue of a non-singular matrix , then an eigenvalue of is:

eigen-values and eigenvectors of a matrix Medium
A.
B.
C.
D.

30 The sum of the eigenvalues of a matrix is equal to:

eigen-values and eigenvectors of a matrix Medium
A. The trace of the matrix
B. Zero
C. The product of its diagonal elements
D. The determinant of the matrix

31 The product of all the eigenvalues of a square matrix is equal to:

eigen-values and eigenvectors of a matrix Medium
A. The trace of
B. The determinant of
C.
D.

32 If a matrix has eigenvalues and , what is the trace and determinant of the matrix, respectively?

eigen-values and eigenvectors of a matrix Medium
A. Trace = , Determinant =
B. Trace = , Determinant =
C. Trace = , Determinant =
D. Trace = , Determinant =

33 Which of the following is true for the eigenvalues of a real symmetric matrix?

eigen-values and eigenvectors of a matrix Medium
A. They are always zero.
B. They are always purely imaginary.
C. They are always real numbers.
D. They are always complex conjugates.

34 The Cayley-Hamilton theorem states that every square matrix satisfies its own:

Cayley-Hamilton theorem Medium
A. Transposed matrix
B. Identity matrix
C. Inverse function
D. Characteristic equation

35 If the characteristic equation of a matrix is , then according to the Cayley-Hamilton theorem:

Cayley-Hamilton theorem Medium
A.
B.
C.
D.

36 Using the characteristic equation obtained from the Cayley-Hamilton theorem, the inverse can be expressed as:

Cayley-Hamilton theorem Medium
A.
B.
C.
D.

37 One of the primary applications of the Cayley-Hamilton theorem is finding:

Cayley-Hamilton theorem Medium
A. The trace of a non-square matrix
B. Higher powers and the inverse of a square matrix
C. The exact eigenvalues of a matrix without solving polynomials
D. The rank of a rectangular matrix

38 If a system of linear equations in variables has a singular coefficient matrix (), the system can be:

inverse of a matrix and solution of linear simultaneous equations Medium
A. Either inconsistent or having infinitely many solutions
B. Only having a trivial solution
C. Only consistent with a unique solution
D. Always inconsistent

39 Which of the following operations is NOT considered a valid elementary row operation?

elementary operations and their use in getting the rank Medium
A. Interchanging two rows
B. Multiplying a row by a non-zero scalar
C. Squaring all the elements of a row
D. Adding a scalar multiple of one row to another row

40 If is an eigenvector corresponding to the eigenvalue of a matrix , then for any non-zero scalar , the vector is:

eigen-values and eigenvectors of a matrix Medium
A. An eigenvector corresponding to the eigenvalue
B. An eigenvector corresponding to the eigenvalue
C. Not an eigenvector
D. An eigenvector corresponding to the eigenvalue

41 Let be a real matrix with eigenvalues and . Which of the following statements about is definitely true?

eigen-values and eigenvectors of a matrix Hard
A. It is similar to a diagonal matrix with diagonal entries .
B. It has eigenvalues .
C. Its trace is .
D. Its eigenvalues are .

42 If , what is the value of the matrix expression ?

Cayley-Hamilton theorem Hard
A.
B.
C.
D.

43 Let be an matrix of rank . If is a matrix obtained by performing a sequence of row operations on , and is formed by replacing the last row of with a linear combination of its other rows, what is the maximum possible rank of ?

elementary operations and their use in getting the rank Hard
A.
B.
C.
D.

44 Consider a system of linear equations where is an matrix () of full row rank. Which of the following is true regarding the solutions to this system?

inverse of a matrix and solution of linear simultaneous equations Hard
A. The solution space forms a subspace of dimension .
B. It has a unique solution for any .
C. It has infinitely many solutions for any .
D. It is inconsistent for some .

45 Let be a nilpotent matrix of index (i.e., ). What is the algebraic multiplicity and geometric multiplicity of the eigenvalue ?

eigen-values and eigenvectors of a matrix Hard
A. Algebraic: 4, Geometric: 1
B. Algebraic: 4, Geometric: 3
C. Algebraic: 3, Geometric: 2
D. Algebraic: 4, Geometric: 2

46 Let be an invertible matrix with characteristic polynomial . What is the trace of in terms of the coefficients ?

Cayley-Hamilton theorem Hard
A.
B.
C.
D.

47 Let be an matrix of rank . Let be the adjugate matrix of . What is the rank of ?

elementary operations and their use in getting the rank Hard
A.
B.
C.
D.

48 For what values of does the system , , have a solution?

inverse of a matrix and solution of linear simultaneous equations Hard
A. For all real
B. or
C. The system has a unique solution for any .
D. Only

49 If is an eigenvector of a matrix corresponding to eigenvalue , and is non-singular, which of the following is an eigenvector of ?

eigen-values and eigenvectors of a matrix Hard
A.
B. It cannot be determined.
C.
D.

50 Given a matrix such that and . What is expressed as a linear combination of and ?

Cayley-Hamilton theorem Hard
A.
B.
C.
D.

51 Let and be two matrices. Which of the following inequalities regarding rank is always true?

elementary operations and their use in getting the rank Hard
A.
B.
C.
D.

52 Let be a matrix such that . Which of the following represents ?

inverse of a matrix and solution of linear simultaneous equations Hard
A.
B.
C.
D.

53 Let be an orthogonal matrix with purely real entries and . Which of the following must be an eigenvalue of ?

eigen-values and eigenvectors of a matrix Hard
A.
B.
C.
D.

54 Let be a matrix with characteristic equation . What is the trace of ?

Cayley-Hamilton theorem Hard
A.
B.
C.
D.

55 Consider a block matrix , where and are square matrices. Which of the following statements about the rank of is always true?

elementary operations and their use in getting the rank Hard
A.
B.
C.
D.

56 Let be a matrix such that the system of equations has a non-trivial solution. Which of the following systems (where ) can have a unique solution?

inverse of a matrix and solution of linear simultaneous equations Hard
A. Only if is in the null space of
B. Only if has no zero entries
C. It is impossible for any
D. Only if is in the column space of

57 If is a real skew-symmetric matrix of odd order , what can be said about its eigenvalues and determinant?

eigen-values and eigenvectors of a matrix Hard
A. It has at least one real non-zero eigenvalue, and .
B. All eigenvalues are real, and .
C. It has 0 as an eigenvalue, and .
D. All eigenvalues are purely imaginary, and .

58 For a matrix with minimal polynomial , what is the inverse of ?

Cayley-Hamilton theorem Hard
A.
B.
C.
D.

59 Let and be non-zero column vectors. What is the rank of the matrix , assuming ?

elementary operations and their use in getting the rank Hard
A. Depends on the values of and
B.
C.
D.

60 If and are non-singular matrices such that is also non-singular, which of the following expressions is equivalent to ?

inverse of a matrix and solution of linear simultaneous equations Hard
A.
B.
C.
D.
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Unit 2