91Ó°ÊÓ

We are given a matrix A such that its determinant is 1: \[ \operatorname{det}(A)=1 \] We want to show that: \[ \operatorname{adj}(\operatorname{adj} A)=A \]

We know that adj(A) can be expressed as the transpose of the cofactor matrix of A, and also the following relationship holds: \[ A \cdot \operatorname{adj}(A) = \operatorname{det}(A) \cdot I \] Where I is the identity matrix. Since \(\operatorname{det}(A)=1\), we have: \[ A \cdot \operatorname{adj}(A) = I \]

We know that if B and C are two matrices, then the adjugate of their product is obtained by reversing the order of their adjugates and taking the product: \[ \operatorname{adj}(B \cdot C) = \operatorname{adj}(C) \cdot \operatorname{adj}(B) \] Let's apply this to the relationships we derived in Step 2. We have \(A \cdot \operatorname{adj}(A) = I\), so we can multiply both sides of the equation by \(\operatorname{adj}(A)\) from the right: \[ A \cdot \operatorname{adj}(A) \cdot \operatorname{adj}(A)= \operatorname{adj}(A) \] Now, we use property 3 and replace \(\operatorname{adj}(A \cdot \operatorname{adj}(A))\) with \(\operatorname{adj}(\operatorname{adj}(A)) \cdot \operatorname{adj}(A)\) to get: \[ \operatorname{adj}(\operatorname{adj}(A)) \cdot \operatorname{adj}(A) = \operatorname{adj}(A) \]

We have: \[ \operatorname{adj}(\operatorname{adj}(A)) \cdot \operatorname{adj}(A) = \operatorname{adj}(A) \] Now, we need to find \(\operatorname{adj}(\operatorname{adj}(A))\). To achieve this, let's multiply both sides of the equation by \(\operatorname{adj}(A)^{-1}\) from the right: \[ \operatorname{adj}(\operatorname{adj}(A)) \cdot \operatorname{adj}(A) \cdot \operatorname{adj}(A)^{-1} = \operatorname{adj}(A) \cdot \operatorname{adj}(A)^{-1} \] Using the fact that \(\operatorname{adj}(A) \cdot \operatorname{adj}(A)^{-1} = I\), we have: \[ \operatorname{adj}(\operatorname{adj}(A)) \cdot I = I \] So, we finally obtain the desired result: \[ \operatorname{adj}(\operatorname{adj}(A)) = A \] We have shown that if \(\operatorname{det}(A)=1\), then \(\operatorname{adj}(\operatorname{adj} A)=A\).