Matrices (1 Viewer)

[tacogym27101990]

hey i just came across a question
and it asked if there is a matrix A to find A^2
and then the 2nd part said find A^-1
and it turned out that A^-1 = -A^2

soooo im just wondering if that was just a coincidence or its a general rule for matrices

thanks

 

[omniscience]

2 x 5 = 10 dude

 

[Iruka]

Not all matrices have an inverse, so it can't be true in general.

If the matrix A is a rotation matrix, i.e., it has the effect of rotating a vector through 2pi/3, however, you can see that A^2=A^-1.

 

[Affinity]

If I remember correctly, the matrices which satisfy A^2 = A^-1 are precisely the ones which are diagonalizable with eigen values which are cube roots of unity

 

[Iruka]

Yeah, I think you can use the Cayley-Hamilton theorem to show something like that.