Question 1

Determine if the given linear transformation 
  is invertible, and if so, find 
  
.

Accepted Answer

The answer of Determine if the given linear transformation 
...

Question 2

If 
  is a square matrix such that 
  has a solution for some particular vector 
  
, then 
  is invertible.

Accepted Answer

The answer of If 
  is a square matrix...

Question 3

If 
  and 
  are 
  invertible matrices, then so is

Accepted Answer

The answer of If 
  and 
  are...

Question 4

If 
  is a square invertible matrix, then 
  is an invertible matrix for every positive integer 
  
, with

Accepted Answer

The given matrix is a square invertible matrix, which means its determinant is non-zero. Therefore, it is guaranteed that its minor matrices are also non-zero. Using the formula for the determinant of a 3x3 matrix, we can see that the determinant of the new matrix can be written as a linear combination of the determinants of its minor matrices, which are non-zero. Thus, the determinant of the new matrix is also non-zero, and hence it is invertible.

Question 5

If 
  is a square invertible matrix, and 
  then

Accepted Answer

The given matrices are related by a single elementary row operation, which is multiplying the second row by -1. This operation does not change the invertibility of the matrix, so the resulting matrix is still a square invertible matrix.

Question 6

If 
  and 
  are 
  matrices, and 
  and 
  are both not invertible, then 
  is not invertible.

Accepted Answer

The answer of If 
  and 
  are...

Question 7

Use the given LU factorization to solve 
  where

Accepted Answer

The answer of Use the given LU factorization to solve...

Question 8

Use the given LU factorization to solve 
  where

Accepted Answer

The answer of Use the given LU factorization to solve...

Question 9

Find an LU factorization of

Accepted Answer

The answer of Find an LU factorization of
 ...

Question 10

Find an LU factorization of

Accepted Answer

The answer of Find an LU factorization of
 ...

Question 11

Find an LU factorization of

Accepted Answer

The answer of Find an LU factorization of
 ...

Question 12

Find an LU factorization of

Accepted Answer

The answer of Find an LU factorization of
 ...

Question 13

Find an LDU factorization of

Accepted Answer

The answer of Find an LDU factorization of
 ...

Question 14

Find an LDU factorization of

Accepted Answer

The answer of Find an LDU factorization of
 ...

Question 15

Determine 
  by evaluating 
  from the LDU factorization

Accepted Answer

The answer of Determine 
  by evaluating 
 ...

Question 16

Every matrix 
  has an LU factorization.

Accepted Answer

The answer of Every matrix 
  has an LU...

Question 17

If 
  is a square invertible matrix which has an LU factorization, then the factorization is unique.

Accepted Answer

Any invertible matrix with an LU factorization has a unique factorization, as long as we require that the diagonal entries of the L factor be 1. This is known as the Doolittle algorithm for LU factorization. Thus, the statement is true.

Question 18

Every matrix 
  has an LDU factorization.

Accepted Answer

The answer of Every matrix 
  has an LDU...

Question 19

If 
  is a square matrix which has an LU factorization, then 
  is invertible.

Accepted Answer

The answer of If 
  is a square matrix...

Question 20

Every elementary matrix has an LU factorization.

Accepted Answer

The answer of Every elementary matrix has an LU factorization....

Quiz 3: Matrices

Determine if the given linear transformation is invertible, and if so, find .

If is a square matrix such that has a solution for some particular vector , then is invertible.

If and are invertible matrices, then so is

If is a square invertible matrix, then is an invertible matrix for every positive integer , with

If is a square invertible matrix, and then

If and are matrices, and and are both not invertible, then is not invertible.

Use the given LU factorization to solve where

Use the given LU factorization to solve where

Find an LU factorization of

Find an LU factorization of

Find an LU factorization of

Find an LU factorization of

Find an LDU factorization of

Find an LDU factorization of

Determine by evaluating from the LDU factorization

Every matrix has an LU factorization.

If is a square invertible matrix which has an LU factorization, then the factorization is unique.

Every matrix has an LDU factorization.

If is a square matrix which has an LU factorization, then is invertible.

Every elementary matrix has an LU factorization.