Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 18 / a / Solution

The middle row is $-1/2$ times the sum of the first and third rows, so the rows are linearly dependent and the determinant is zero. The matrix trace is the sum of diagonal entries, $1+4+1=6$. A transpose preserves the determinant, while the determinant of a product is the product of its determinants. Consequently