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Unit 3: Linear Algebra - Practice Quiz

CSE333 — Combinatorial Studies-I 60 Questions
0 Correct 0 Wrong 60 Left
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1 What is the order of the matrix ?

matrices Easy
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B.
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2 Which matrix has all entries equal to zero?

matrices Easy
A. Scalar matrix
B. Identity matrix
C. Zero matrix
D. Diagonal matrix

3 What is the transpose of ?

matrices Easy
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4 Which matrix is the identity matrix?

matrices Easy
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5 What is the determinant of ?

determinants Easy
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6 What is the determinant of the identity matrix ?

determinants Easy
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7 If a square matrix has two identical rows, what is its determinant?

determinants Easy
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8 A square matrix is invertible when which condition holds?

determinants Easy
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9 Which ordered pair solves the system and ?

system of linear equations Easy
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10 What does a consistent system of linear equations have?

system of linear equations Easy
A. Only the zero solution
B. No possible solution
C. Exactly two solutions
D. At least one solution

11 Which matrix form represents the linear system ?

system of linear equations Easy
A. Variable vector times constant vector equals coefficient matrix
B. Constant matrix plus variable vector equals coefficient vector
C. Coefficient vector minus constant matrix equals variable matrix
D. Coefficient matrix times variable vector equals constant vector

12 If two equations represent the same straight line, how many solutions does the system have?

system of linear equations Easy
A. No solutions
B. Infinitely many solutions
C. Exactly two solutions
D. Exactly one solution

13 If for a nonzero vector , what is called?

eigenvalues Easy
A. A transpose of
B. A determinant of
C. An inverse of
D. An eigenvalue of

14 What equation is used to find the eigenvalues of a square matrix ?

eigenvalues Easy
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15 What are the eigenvalues of the diagonal matrix ?

eigenvalues Easy
A. and
B. and
C. and
D. and

16 Which vector cannot be an eigenvector?

eigenvectors Easy
A. A unit vector
B. The zero vector
C. A nonzero vector
D. A column vector

17 For , which vector is an eigenvector corresponding to ?

eigenvectors Easy
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18 If is an eigenvector of with eigenvalue , which equality is true?

eigenvectors Easy
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19 In LU decomposition, a matrix is written in which form?

LU decomposition Easy
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20 In the factorization , what type of matrix is ?

LU decomposition Easy
A. Lower triangular matrix
B. Upper triangular matrix
C. Diagonal zero matrix
D. Symmetric identity matrix

21 Let and . What is ?

matrices Medium
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22 If , what is ?

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23 Let and . What is the entry in row , column of ?

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24 Evaluate .

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25 Suppose . Matrix is obtained by interchanging two rows of and then multiplying one row by . What is ?

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26 For which values of is the matrix singular?

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27 If is a matrix with , what is ?

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28 Solve the system , , and .

system of linear equations Medium
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29 For what value of does the system and have infinitely many solutions?

system of linear equations Medium
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30 A system reduces to and , with free. Which expression describes its solution set?

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31 For which values of does the system and have a unique solution?

system of linear equations Medium
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32 What are the eigenvalues of ?

eigenvalues Medium
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33 Find the eigenvalues of .

eigenvalues Medium
A. and
B. and
C. and
D. and

34 A matrix has eigenvalues , , and . What are the eigenvalues of ?

eigenvalues Medium
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35 Which vector is an eigenvector of corresponding to the eigenvalue ?

eigenvectors Medium
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36 Which vector is an eigenvector of corresponding to the eigenvalue ?

eigenvectors Medium
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37 For , which matrix diagonalizes as ?

eigenvectors Medium
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38 Using Doolittle's method, which is the LU decomposition of ?

LU decomposition Medium
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39 Suppose , where and . What is the solution of ?

LU decomposition Medium
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40 Using Gaussian elimination without row exchanges, what is the lower triangular factor in the LU decomposition of ?

LU decomposition Medium
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41 A real matrix satisfies and . What is ?

matrices Hard
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C. The determinant cannot be determined because the matrix entries and eigenvectors are unspecified.
D.

42 Let be an invertible matrix with . If column vectors satisfy , find .

determinants Hard
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B.
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43 For which values of and does the system

have no solution?

system of linear equations Hard
A. and
B. and
C. and
D. and

44 A matrix has characteristic polynomial and minimal polynomial . What is ?

eigenvectors Hard
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B.
C.
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45 For which real values of does

admit an LU decomposition without row exchanges, with all diagonal entries of equal to ?

LU decomposition Hard
A.
B.
C. , because only the second leading principal minor controls the existence of such a decomposition
D.

46 Evaluate

determinants Hard
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B.
C.
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47 For a real matrix , which statement about the matrix equation is correct?

matrices Hard
A. It has a unique solution whenever is invertible.
B. It is solvable precisely when is diagonalizable over the real numbers and has no repeated eigenvalues.
C. It has infinitely many solutions whenever has distinct eigenvalues.
D. It has no solution for any .

48 Consider , where
Which statement is correct?

system of linear equations Hard
A. For , the system has a unique solution.
B. For every , the system has infinitely many solutions.
C. For , the solution set has dimension .
D. For , the solution set has dimension .

49 Let be an invertible real matrix satisfying . Which value cannot equal ?

eigenvalues Hard
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B.
C.
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50 Let . Which vector is an eigenvector of but not an eigenvector of ?

eigenvectors Hard
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C.
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51 Using Doolittle factorization , where has unit diagonal, let
What is the ordered pair ?

LU decomposition Hard
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C.
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52 Let be a real skew-symmetric matrix with . What is ?

determinants Hard
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53 Let be a matrix of rank with and . If and is one solution of , what is the complete solution set?

system of linear equations Hard
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54 A matrix has eigenvalues , , and . Find

eigenvalues Hard
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55 Suppose is an complex matrix with distinct eigenvalues and . Which conclusion must hold?

eigenvectors Hard
A. Every eigenvector of is an eigenvector of .
B. must be a polynomial in of degree exactly , with a nonzero leading coefficient.
C. and must have exactly the same distinct eigenvalues.
D. Every eigenvector of is an eigenvector of .

56 If and are matrices with , what is the smallest possible value of ?

matrices Hard
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57 For an arbitrary matrix , evaluate the block determinant

determinants Hard
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58 Let be the tridiagonal matrix with every diagonal entry equal to and every entry immediately above or below the diagonal equal to . In the Doolittle factorization , what is ?

LU decomposition Hard
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C.
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59 Let be invertible and let nonzero vectors satisfy . For the system , which statement is correct?

system of linear equations Hard
A. It is solvable exactly when , and then it has a unique solution.
B. It is solvable exactly when , and then it has a one-parameter family of solutions.
C. It has a unique solution for every because adding a rank-one matrix to an invertible matrix preserves invertibility.
D. It is solvable exactly when , and then it has a one-parameter family of solutions.

60 A matrix satisfies , , and . What is the spectral radius of ?

eigenvalues Hard
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B.
C.
D.
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