Let M be a 3 x 3 non-singular matrix with det(M) = 4. If M<sup>-1</sup> adj(adj M) = k<sup>2</sup>I, then the value of 'k' may be :
Step-by-Step Solution
Key Concept: Use the property adj(adj M) = |M|^(n-2) * M, where n is the order of the matrix. Here n=3, so adj(adj M) = |M| * M. Then M^-1 * adj(adj M) = M^-1 * |M| * M = |M| * I. Given |M|=4, we have 4I = k^2I, so k^2 = 4, which means k = 2 or -2.
Given M is a 3x3 matrix, n=3. We know that adj(adj M) = |M|^(n-2) * M. Since n=3, adj(adj M) = |M|^(3-2) * M = |M| * M. Given |M| = 4, we have adj(adj M) = 4M. Now, M^-1 * adj(adj M) = M^-1 * (4M) = 4 * (M^-1 * M) = 4I. We are given M^-1 * adj(adj M) = k^2I. Therefore, k^2I = 4I, which implies k^2 = 4. Thus, k = 2 or k = -2. Looking at the options, +2 is present.
Correct Answer: 1