**Invertible and Elementary Matrices Question 1**

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

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

Matrix Inversion, Elementary matrices

**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

**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?)

**Invertible and Elementary Matrices**