Divison in a finite field

Question

I was working on a linear algebra problem in F121 (i.e the finite field with 121 elements) in which I had to multiply a matrix by 1/4. The answer is shown in the picture. I'm am very confused to how this was reached. Any help is appreciated matrix

Answer 1

The field $F_{121}$ is a quadratic extension of $F_{11}.$ The elements of $F_{121}$ are usually represented as polynomials of degree less than two with coefficients in $F_{11}$ but in this case, all the elements we see are constants. That means that we are actually working in $F_{11},$ so it's just arithmetic modulo $11.$ Yes, $1/4=3,$ and for a problem like this, brute force is as good as anything else.

Now just do the arithmetic modulo $11,$ and you'll get the answer shown.