
theorem
  for N being Matrix of 3,F_Real st
  N = <* <* 2,   0,    -1 *>,
         <* 0, sqrt 3,  0 *>,
         <* 1,   0,    -2 *> *> holds
  Det N = (-3) * sqrt 3 & N is invertible
  proof
    let N be Matrix of 3,F_Real;
    assume
A1: N = <* <* 2,   0   ,-1 *>,
           <* 0, sqrt 3, 0 *>,
           <* 1,   0   ,-2 *> *>;
    reconsider a = 2,b = 0,c = -1,d = 0,e = sqrt 3,f = 0,g = 1,h = 0,i = -2
      as Element of F_Real by XREAL_0:def 1;
    Det N = a*e*i - c*e*g - a*f*h + b*f*g - b*d*i + c*d*h by A1,MATRIX_9:46;
    then Det N = (-3) * sqrt 3 & Det N <> 0.F_Real by SQUARE_1:24;
    hence thesis by LAPLACE:34;
  end;
