linear algebra

Invertible and Elementary Matrices

August 7, 2020

Question 1

For the following, give an example if one exists, or explain why no such example exists.

a) A 3x3 matrix which has a nontrivial null space. b) An invertible 4x4 matrix whose columns do not span R4 c) An invertible 3x3 matrix A, along with two 3x3 matrices B,C such that AB=AC but B 6=C d) Two nonzero 3x3 matrices A,B such that AB=03x3=BA (where 03x3 is the 3x3 matrix of all 0’s)

Question 2

Determine if the following matrices are invertible. If they are invertible find their inverse.

a)A=

( 2 3 4 5

)

b) A=

 1 0 00 2 1

1 0 1

 .

c) A=

 1 0 12 1 3

3 0 3

 .

1

Question 3

Consider the matrix A=

 1 4 72 5 8

3 6 9

 . First, compute the following three

matrix multiplications

A

 1 0 00 0 1

0 1 0

  (1)

A

 1 0 00 1 0

0 0 4

  (2)

A

  1 0 0−3 1 0

0 0 1

  (3)

State how these three matrices you get after computing the multiplication are related to the original matrix A. Is there a pattern, and can a general result be conjectured from this? (Hint, the matrices you are asked to multiply A by are elementary matrices: what happens when you multiply a matrix by an elementary matrix on the left?)

2