# 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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