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