Using properties of determinants prove that: [(a^2 + 2a, 2a+1, 1)(2a+1, a+2,1)(3,3,1)] = (a-1)^3

Question

Using properties of determinants prove that:

\(\begin{bmatrix} a^2 + 2a & 2a + 1 & 1 \\[0.3em] 2a + 1 & a+2 & 1 \\[0.3em] 3 & 3 & 1 \end{bmatrix}\) = (a-1)³

Answer 1

\(\begin{bmatrix} a^2 + 2a & 2a + 1 & 1 \\[0.3em] 2a + 1 & a+2 & 1 \\[0.3em] 3 & 3 & 1 \end{bmatrix}\)

= (a - 1)²[a + 1 - 0 - 2] [expansion by first row] 

= (a - 1)³