Algebra homework help

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 3×3 matrix which has a nontrivial null space.
b) An invertible 4×4 matrix whose columns do not span R
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c) An invertible 3×3 matrix A, along with two 3×3 matrices B,C such that
AB=AC but B6=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)
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 0
0 2 1
1 0 1

.
c) A=


1 0 1
2 1 3
3 0 3

.
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Question 3
Consider the matrix A=


1 4 7
2 5 8
3 6 9

. First, compute the following three
matrix multiplications
A


1 0 0
0 0 1
0 1 0

 (1)
A


1 0 0
0 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?)
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