Question 1

If both A and B are 
  matrices such that their product AB has an LU factorization, then both A and B have LU factorizations.

Accepted Answer

The answer of If both A and B are 
...

Question 2

Determine if the matrix is a stochastic matrix.

Accepted Answer

The answer of Determine if the matrix is a stochastic...

Question 3

Determine if the matrix is a stochastic matrix.

Accepted Answer

The answer of Determine if the matrix is a stochastic...

Question 4

Find the state vector 
  for the given stochastic matrix and initial state vector.

Accepted Answer

The answer of Find the state vector 
  for...

Question 5

Find the state vector 
  for the given stochastic matrix and initial state vector.

Accepted Answer

The answer of Find the state vector 
  for...

Question 6

Find all steady-state vectors for the given stochastic matrix.

Accepted Answer

The answer of Find all steady-state vectors for the given...

Question 7

Find all steady-state vectors for the given stochastic matrix.

Accepted Answer

The answer of Find all steady-state vectors for the given...

Question 8

Find all steady-state vectors for the given stochastic matrix.

Accepted Answer

The answer of Find all steady-state vectors for the given...

Question 9

Find all steady-state vectors for the given stochastic matrix.

Accepted Answer

The answer of Find all steady-state vectors for the given...

Question 10

Determine if the given matrix is regular.

Accepted Answer

The answer of Determine if the given matrix is regular.
...

Question 11

Determine if the given matrix is regular.

Accepted Answer

The answer of Determine if the given matrix is regular.
...

Question 12

If 
  and 
  are 
  stochastic matrices, then 
  is a stochastic matrix.

Accepted Answer

The answer of If 
  and 
  are...

Question 13

A matrix 
  is doubly stochastic if and only if 
  is doubly stochastic.

Accepted Answer

A matrix is said to be doubly stochastic if the sum of elements in each row as well as each column is equal to 1. It can be easily shown that if a matrix satisfies this property, then its transpose also satisfies the property. Therefore, if the given matrix is doubly stochastic, its transpose (which is the matrix in the second part of the statement) will also be doubly stochastic, and vice versa. Hence, the answer is A, true.

Question 14

The 
  identity matrix 
  is a regular stochastic matrix.

Accepted Answer

The identity matrix is a special kind of matrix where the diagonal elements are 1 and all other elements are 0. It is not a stochastic matrix as the columns do not add up to 1. A regular stochastic matrix is a matrix where all elements are nonnegative and each column adds up to 1.

Question 15

If 
  and 
  are 
  doubly stochastic matrices, then AB is a doubly stochastic matrix.

Accepted Answer

The product of two doubly stochastic matrices is also a doubly stochastic matrix.

Question 16

If 
  and 
  are 
  regular stochastic matrices, then AB is a regular stochastic matrix.

Accepted Answer

The answer of If 
  and 
  are...

Quiz 3: Matrices

If both A and B are matrices such that their product AB has an LU factorization, then both A and B have LU factorizations.

Determine if the matrix is a stochastic matrix.

Determine if the matrix is a stochastic matrix.

Find the state vector for the given stochastic matrix and initial state vector.

Find the state vector for the given stochastic matrix and initial state vector.

Find all steady-state vectors for the given stochastic matrix.

Find all steady-state vectors for the given stochastic matrix.

Find all steady-state vectors for the given stochastic matrix.

Find all steady-state vectors for the given stochastic matrix.

Determine if the given matrix is regular.

Determine if the given matrix is regular.

If and are stochastic matrices, then is a stochastic matrix.

A matrix is doubly stochastic if and only if is doubly stochastic.

The identity matrix is a regular stochastic matrix.

If and are doubly stochastic matrices, then AB is a doubly stochastic matrix.

If and are regular stochastic matrices, then AB is a regular stochastic matrix.