Math 2270
Quiz 3
1. Decide whether the following matrix is invertible and find the inverse if it exists. If you claim the inverse does
not exist, tell me why this is the case.
2 3
6 9
Recall that we call a matrix A invertible if we can solve the equation A~x = ~b uniquely for any choice of ~b. Let’s
see if that is the case here.

 

..
..
b1
3
1
.
2
3
.
b
1 
2
2
.

∼
..
..
0 0 . b − 3b
6 9 . b
2
2
1
From here we can see that the equation A~x = ~b cannot always be solved, in particular when the components of ~b
satisfy b2 − 3b1 6= 0. Therefore, the matrix in question is not invertivble.
2. Consider a matrix of the form
A=
a
b
b
−a
,
where a2 + b2 = 1 (a reflection about some line L). Find all nonzero vectors ~v such that A~v = ~v in terms of ~u,
where ~u is a unit vector parallel to L.
The matrix A represents a reflection about a line, and we are looking for a vector that A holds fixed. Obviously,
and vector not on the line will be reflected about the line by A and therefore will not be fixed by the transformation.
However, if ~v is a vector on L, then its reflection about L is itself, so it is fixed by the transformation. In particular,
we have ~v = k~u for some scalar k.