Question:

Step 1: Rewrite the given expression by grouping terms to identify perfect squares. $a^2 + 8b^2 + 2c^2 + 2d^2 - 4ab - 4bc - 4bd = 0$. Step 2: Group as sum of squares: $(a - 2b)^2 + 2(c - b)^2 + 2(d - b)^2 = 0$. Verify: $(a-2b)^2 = a^2 - 4ab + 4b^2$; $2(c-b)^2 = 2c^2 - 4bc + 2b^2$; $2(d-b)^2 = 2d^2 - 4bd + 2b^2$. Sum $= a^2 + 8b^2 + 2c^2 + 2d^2 - 4ab - 4bc - 4bd$. ✓ Step 3: Since each squared term is non-negative and their sum is zero, each must be zero: $a - 2b = 0 \Rightarrow a = 2b$; $c - b = 0 \Rightarrow c = b$; $d - b = 0 \Rightarrow d = b$. Step 4: Compute the determinant: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc = (2b)(b) - (b)(b) = 2b^2 - b^2 = b^2$. Step 5: Express in terms of $a$: since $a = 2b$, $b = \frac{a}{2}$, so $b^2 = \frac{a^2}{4}$. Thus the determinant equals both $\frac{a^2}{4}$ and $b^2$.