Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M^{-1} = \operatorname{adj} (\operatorname{adj} M)$, then which of the following statements is/are ALWAYS TRUE?

Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M⁻¹ = adj (adj M), then which of the following statements is/are ALWAYS TRUE? $M_{3\times 3} \text{ invertible},\ \ I = 3\times 3 \text{ identity}$ $M^{-1} = \operatorname{adj}(\operatorname{adj} M)$ $\text{Check (1), (2), (3), (4) — which are ALWAYS true?}$ $\text{use } \operatorname{adj}(\operatorname{adj} A) = (\det A)^{n-2} A$ $1.\ \operatorname{adj}(\operatorname{adj} A) = (\det A)^{n-2} A$ $2.\ \det \Rightarrow \det M$ $3.\ \text{check options}$ $\operatorname{adj}(\operatorname{adj} A) = (\det A)^{n-2}A;\ \ n=3 \Rightarrow (\det M)\,M$ $M^{-1} = (\det M)\,M$ $\det(M^{-1}) = \det((\det M)M) \Rightarrow \frac{1}{\det M} = (\det M)^3\det M$ $(\det M)^5 = 1 \Rightarrow \det M = 1\ (\text{real})$ $\det M = 1 \Rightarrow (2)\ \text{TRUE}$ $M^{-1} = M$ $M^{-1} = M \Rightarrow M^2 = I$ $(3)\ \text{TRUE}$ $M = \operatorname{diag}(1,-1,-1):\ M^2=I,\ \det M=1,\ M\neq I$ $(1)\ \text{FALSE}$ $\operatorname{adj} M = \det(M)\,M^{-1} = M$ $(\operatorname{adj} M)^2 = M^2 = I \Rightarrow (4)\ \text{TRUE}$ $\text{True: (2), (3), (4)}$ $\boxed{\text{Answer: (2), (3), (4)}}$