Find all 2x2 matrices

Question

Find all 2x2 matrices A that are symmetric and whose squares are equal to themselves

Answer 1

To me, the most obvious answers come from

 

Let M be the symmetric matrix

 

[a a]

[a a]

 

If M^2 = M, then

 

a = 2a^2

0 = a(2a -1)

 

so the zeroes are 0 and 1/2, giving

 

[0 0]

[0 0]

 

and

 

[1/2 1/2]

[1/2 1/2]

 

The second one was one of Francisco's examples, but the first was missed by his formulae due to dividing his second initial equation by C to show that (a+b) = 1.

 

So, basically, his argument correctly gives the family of answers where c is not equal to 0, and needs to have a second family of answers where C = 0.

 

c = 0

a^2 = a

b^2=b

 

Giving

 

[0 0]

[0 0] 

 

[1 0]
[0 0] 

 

[0 0]
[0 1]

 

[1 0]
[0 1] 

 

 

Answer 2

Let a and b be the diagonal entries and c be the off-diagonal entries.

 

 

By multiplication,

 

 a² + c² = a

 

c(a + b) = c

 

b² + c² = b

 

 

From the equations above,

 

a + b = 1

 

a² + b² + 2c² = 1

 

 

 

In terms of a,

 

b = 1 - a

 

c = ±√[½(1 - a² - (1 - a)²] =± √[a(1 - a)]