Question

If \( A \) is an invertible matrix of order \( n \), then \( \text{adj} (\text{adj} A) = |A|^{n-2} A \).

Hint:

For an invertible matrix \( A \), \( \text{adj}(\text{adj}(A)) = |A|^{n-2} A \), which is useful in matrix theory and properties of adjugates.

Answer 1

Step 1: Understanding the adjugate matrix. 
The adjugate (or adjoint) of a matrix \( A \), denoted \( \text{adj}(A) \), is the transpose of the cofactor matrix of \( A \). The property of the adjugate matrix for an invertible matrix \( A \) is given by: \[ A \cdot \text{adj}(A) = |A| \cdot I \] where \( I \) is the identity matrix. 
Step 2: Applying the property of the adjugate. 
For the adjugate of the adjugate, we have the relation: \[ \text{adj}(\text{adj}(A)) = |A|^{n-2} A \] This is a known property of the adjugate matrix, where \( n \) is the order of the matrix. 
Conclusion: 
Thus, the statement is True.