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

Apply the elementary row operation $R_1\leftarrow R_1-R_2$, which preserves the determinant. Expanding along the new first row givesA square matrix is an invertible matrix exactly when its determinant is nonzero. Thereforeare precisely the three values making $A$ a singular matrix. They also have direct explanations: at $c=0$ the second row is zero, at $c=2$ the first two rows coincide, and at $c=7$ the last two columns coincide.

For either an upper triangular matrix or a lower triangular matrix,In the permutation formula for the determinant, every contribution except the diagonal product includes an entry on the zero side of the diagonal. Equivalently, successive cofactor expansions along the first column of an upper triangular matrix prove the formula by mathematical induction.

For $i\ne j$, the elementary row operation $R_i\leftarrow R_i-k{R}_j$ leaves the determinant unchanged:By multilinearity in row $i$, the left side is $\det A$ minus $k$ times a determinant with row $j$ appearing twice. That second determinant is zero by alternation. No row scaling or row interchange is involved.

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 $t=\pm 1$.

The elementary row operations preserve the determinant, and the diagonal product of the upper triangular matrix found above isConsequentlyAt $t=1$, all four rows are equal. At $t=-1$, successive rows alternate between one row and its negative. In either case the matrix rank is one, confirming the vanishing determinant.