Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Digital systems, linear algebra, introduction to formal proof / 4 / d / Solution

Perform the elementary row operations from the bottom upwards:This order matters because each subtraction uses the predecessor's original row. For example,The other two differences are $(0,0,1-t^2,t-t^3)$ and $(0,1-t^2,t-t^3,t^2-t^4)$. Thus the required upper triangular matrix isThese elementary row operations use no division, so the reduction remains valid for all real numbers $t$, including $t=\pm 1$.