# The set of matrices that are their own inverse in R2

• Elwin.Martin
In summary, the task is to find all 2x2 square matrices A which are their own inverses. The key equations to use are A2=I and A=A-1. After some attempts at solving algebraically, it was realized that the solution will have diagonals of (+/-)1 and the remaining entries as 0. However, this is not the complete solution. By substituting a=-d and using the equation a2+bc=1, it is possible to express b in terms of a and c. This leads to a more accurate and complete solution.
Elwin.Martin

## Homework Statement

Find all 2x2 square matrices A which are their own inverses.

A2=I
A=A-1

## The Attempt at a Solution

I know that the diagonal is comprised of 1s and or -1s and the other entries are zero but I can't seem to show it algebraically.

I went the weak way and did components...
I said A had rows of {a,b} and {c,d} and that A2=I
so
a2+bc=1
b(a+d)=0
c(a+d)=0
d2+bc=1

Now I know that the solution will have diagonals comprised of (+ or -)1 and the other two entries zero but I don't know how to do the algebra to get there =| I'll write down what I tried last night, any direction would be greatly appreciated!

So I said that
b(a+d)=0
c(a+d)=0
so either (a+d)=0 or b=0 and c=0...
so if we assume a+d=!0 then
b=0 and c=0
thus d2=1
and a2=1
so we'd have a=+-1,b=0,c=0,d=+-1

but if a+d=0
then a=-d
d2-a2=0
d=+-a but apparently the positive case is ignored since we assumed a+d=0 s0 d=-a again and I have nothing conclusive about b or c?

Again, thank you in advanced for any and all advice...it's sad that my basic algebra skills are so weak...if someone could recommend a good algebra practice book that would also be appreciated. I think I'm worse at Algebra than everything else haha

Also if this thread needs to be moved just let me know, at my university Linear Algebra is beyond Calculus but this particular question is fairly simple.

Thank you again for your time
elwin

Elwin.Martin said:

## Homework Statement

Find all 2x2 square matrices A which are their own inverses.

A2=I
A=A-1

## The Attempt at a Solution

I know that the diagonal is comprised of 1s and or -1s and the other entries are zero but I can't seem to show it algebraically.

I went the weak way and did components...
I said A had rows of {a,b} and {c,d} and that A2=I
so
a2+bc=1
b(a+d)=0
c(a+d)=0
d2+bc=1

Now I know that the solution will have diagonals comprised of (+ or -)1 and the other two entries zero but I don't know how to do the algebra to get there =|

Why do you think these are all the matrices? This isn't true.

I'll write down what I tried last night, any direction would be greatly appreciated!

So I said that
b(a+d)=0
c(a+d)=0
so either (a+d)=0 or b=0 and c=0...
so if we assume a+d=!0 then
b=0 and c=0
thus d2=1
and a2=1
so we'd have a=+-1,b=0,c=0,d=+-1

but if a+d=0
then a=-d
d2-a2=0
d=+-a but apparently the positive case is ignored since we assumed a+d=0 s0 d=-a again and I have nothing conclusive about b or c?

So you know you have a=-d. And you also know that a2+bc=1. So you can express b in terms of a and c...

micromass said:
Why do you think these are all the matrices? This isn't true.
Oh wow...that changes things quite a bit...

micromass said:
So you know you have a=-d. And you also know that a2+bc=1. So you can express b in terms of a and c...

Thank you very much! I think I know what I'm doing wrong now.
elwin

## 1. What is the set of matrices that are their own inverse in R2?

The set of matrices that are their own inverse in R2 is a collection of all 2x2 matrices where the product of a matrix and its inverse results in the identity matrix. In other words, each matrix in this set is its own inverse.

## 2. How do you determine if a matrix is its own inverse in R2?

To determine if a matrix is its own inverse in R2, you can simply multiply the matrix by itself and see if the product results in the identity matrix. If it does, then the matrix is its own inverse.

## 3. How many matrices are there in the set of matrices that are their own inverse in R2?

There are an infinite number of matrices in the set of matrices that are their own inverse in R2. This is because for every unique matrix, you can find another matrix that is its own inverse.

## 4. What is the significance of matrices that are their own inverse in R2?

Matrices that are their own inverse in R2 are important in linear algebra as they have many useful properties. These matrices can be used to easily solve systems of linear equations, and they also play a crucial role in the theory of rotations and reflections in geometry.

## 5. Can a matrix be its own inverse in R2 but not in R3 or higher dimensions?

Yes, a matrix can be its own inverse in R2 but not in R3 or higher dimensions. This is because in higher dimensions, a matrix would need to have more than just 2 columns and rows to be its own inverse. In this case, the matrix would no longer be considered to be part of the set of matrices that are their own inverse in R2.

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