Chapter 1.5

Question 0

An elementary matrix

Answer 0

An nxn matrix that one can obtain from the nxn identity matrix I(n) by performing a single elementary row operation

Question 1

Def: if either Matrices A or B can be obtained from the other by a sequence of elementary row operations, they are said to be

Answer 1

Row equivalent

Question 2

Theorem:

Row operations by Matrix multiplication

Answer 2

If the elementary matrix E results from a certain row operation on I(m) and if A is an mxn matrix, then the product EA is the result when this same operation is performed on A.

Question 3

What can be said about theinverse of elementary matrices

Answer 3

Elementary matrices are invertible and the inverse is also an elementary matrix

Question 4

Equivalent statements 1 (4)

If A is an invertible matrix then

Answer 4

1) A is a square matrix
2) Ax=0 only has the trivial solution
3) The reduced row echelon form of A is the identity matrix
4) A is expresseble as a product of elementary matrices

Question 5

What is the Inverse Algorithm

Answer 5

To find the inverse of an invertable matrix A, find a sequence of elementary row operations that reduce A to the Identity matrix and then perform the same sequence of operations on the identity matrix to obtain the inverse of A

Question 6

If A is a singular square matrix, what can be said about the homogeneous linear system Ax=0?

Answer 6

It has infinitally many solutions