Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M<sup>-1</sup> = adj(adj M), then which of the following statement is/are ALWAYS TRUE?
Step-by-Step Solution
Key Concept: Use the property adj(adj M) = (det M)^(n-2) * M, where n=3. So adj(adj M) = (det M) * M. Given M^-1 = (det M) * M, multiply by M to get I = (det M) * M^2. Taking determinant on both sides, 1 = (det M)^3 * (det M)^2 = (det M)^5, so det M = 1. Then M^2 = I.
We know that adj(adj M) = (det M)<sup>n-2</sup> M. For n=3, adj(adj M) = (det M) M. Given M<sup>-1</sup> = adj(adj M), we have M<sup>-1</sup> = (det M) M. Multiplying by M, we get I = (det M) M<sup>2</sup>. Taking determinant on both sides, det(I) = det((det M) M<sup>2</sup>) ⇒ 1 = (det M)<sup>3</sup> (det M)<sup>2</sup> = (det M)<sup>5</sup>. Thus, det M = 1. Substituting det M = 1 into I = (det M) M<sup>2</sup>, we get M<sup>2</sup> = I. Since M<sup>2</sup> = I, (adj M)<sup>2</sup> = (det M)<sup>2</sup> (M<sup>-1</sup>)<sup>2</sup> = 1 * (M<sup>2</sup>)<sup>-1</sup> = I<sup>-1</sup> = I. Thus, (A), (B), (C), and (D) are all correct.
Correct Answer: 1, 2, 3