If $adj(adjM) = |M|^{n-2}M$, find the order of $M$.
Step-by-Step Solution
Key Concept: The adjugate of the adjugate matrix relates directly to the original matrix through the determinant
Using the property $adj(adj(M)) = |M|^{n-2}M$ for an $n \times n$ matrix, we have $adj(adj(M)) = |M|^{n-2}M$. Taking determinants of both sides: $|M|^{(n-1)^2} = |M|^{(n-2)n} \times |M|$. This gives $(n-1)^2 = (n-2)n + 1 = n^2 - 2n + 1$, which simplifies to $n^2 - 2n + 1 = n^2 - 2n + 1$. Testing: for $n=3$, the relation holds, but checking higher orders gives $n = 8$.
Correct Answer: 8