Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Digital systems, linear algebra, introduction to formal proof / 4 / a / Solution

Apply the elementary row operations from bottom to top: $R_4\leftarrow R_4-R_3$, $R_3\leftarrow R_3-R_2$, and $R_2\leftarrow R_2-R_1$. This order ensures that each subtraction uses the original preceding row. The result is the upper triangular matrixPut ${\delta }_1=a$, ${\delta }_2=b-a$, ${\delta }_3=c-b$, ${\delta }_4=d-c$. Row $i$ is either zero if ${\delta }_i=0$, or has its first nonzero entry in column $i$. Thus, for arbitrary real parameters, a row echelon form is obtained by retaining the nonzero rows of $T$ in their existing order and moving every zero row to the bottom. This deals explicitly with vanishing parameters; an upper triangular matrix with an intermediate zero row is not yet a row echelon form.

The subtractions preserve the determinant, so the original matrix hasA square matrix is invertible exactly when its determinant is nonzero. Consequently,These conditions concern consecutive parameters: no ordering, positivity, or pairwise distinctness of all four is required. More generally the rank of a matrix here is the number of nonzero ${\delta }_i$, because the retained nonzero rows have distinct leading columns.