Chapter 1: Linear Systems

Question 1

What is elementary row operation #1?

Answer 1

Add a scalar multiple of one row to another row

Question 2

A non-square matrix can be regular

Answer 2

False

Question 3

Regular matrix can be reduced to UT form without row interchanges.

Answer 3

True

Question 4

The diagonal entries of a regular matrix must be non-zero

Answer 4

True

Question 5

Not all elementary matrices are invertible

Answer 5

False

Question 6

Suppose E is an elementary matrix of type #1. It has a left inverse matrix L which reverts its action. Describe L.

Answer 6

L is a lower unitriangular matrix with the same entries as E but with opposite sign.

Question 7

A matrix A is regular iff it can be factorized as A = LU where L is a lower uni-triangular matrix and U is an upper triangular matrix with non-zero entries

Answer 7

True

Question 8

If A = LU, then L and U are unique

Answer 8

True

Question 9

Every non-singular matrix is regular

Answer 9

False

Question 10

Suppose P is a fixed permutation matrix and PA = LU. Is PA regular?

Answer 10

Yes

Question 11

Suppose PA = LU. Are L and U unique?

Answer 11

No (since P is not unique, the factorization is not unique either)

Question 12

Every regular matrix is non-singular

Answer 12

True

Question 13

A matrix is non-singular iff it has a PA = LU factorization where P is a permutation matrix, L is a lower uni-triangular matrix and U is an upper triangular matrix with non-zero diagonal entries.

Answer 13

True

Question 14

For linear operators, left inverse = right inverse

Answer 14

True

Question 15

A matrix is regular iff it has an LDV factorization, where L is lower uni-tri, U is upper uni-tri and D is diagonal with non-zero entries

Answer 15

True

Question 16

A matrix is invertible iff there exists a permutation matrix P such that PA = LDV.

Answer 16

True

Question 17

Suppose there exists a P such that PA = LDV. Are L D V unique?

Answer 17

No

Question 18

A is regular iff A has a unique A = LU factorization

Answer 18

True

Question 19

A is invertible iff PA has a (possibly non-unique) LU factorization

Answer 19

True

Question 20

A symmetric matrix is regular iff it can be factored as A = LDL^T, where L is lower uni tri and D is diagonal with non-zero entries

Answer 20

True

Question 21

Every symmetric matrix has an LDL^T factorization

Answer 21

False

Question 22

A symmetric matrix is invertible iff there exists a P such that PA has an LDL^T factorization

Answer 22

True