If $K = 4\Delta^3$, where $\Delta$ is the determinant of a $3 \times 3$ matrix, find $K$.
Step-by-Step Solution
Key Concept: Row operations and determinant properties allow us to simplify and compute determinant expressions systematically.
From the given context, $\Delta = 4\Delta^3$ implies we need to solve for $\Delta$. If $\Delta = 4\Delta^3$, then $4\Delta^3 - \Delta = 0$, so $\Delta(4\Delta^2 - 1) = 0$. This gives $\Delta = 0$ or $\Delta^2 = 1/4$. Based on the pattern shown in Q9 where $\Delta = 5$, we can verify the relationship. The answer $K = 4\Delta^3$ evaluated at the operations shown yields $K = 4$.
Correct Answer: 4