What are the properties of the adjoint of a matrix?

Steps to Solve

1. Understanding Adjoint Properties
   The properties of the adjoint of a matrix relate the adjoint to determinants and the original matrix. The given properties are critical in linear algebra.

2. Property 1: Multiplication with Adjoint
   For any matrix ( A ), it holds that: $$ A \cdot (\text{adj}(A)) = (\text{adj}(A)) \cdot A = |A| I $$
   This means multiplying the matrix ( A ) by its adjoint gives a scalar multiple of the identity matrix, where ( |A| ) is the determinant of ( A ).

3. Property 2: Determinant of the Adjoint
   The determinant of the adjoint of ( A ) can be expressed as: $$ |\text{adj}(A)| = |A|^{n-1} $$
   where ( n ) is the order (size) of the square matrix ( A ).

4. Property 3: Determinant in Product with Adjoint
   The determinant of the product of ( A ) and its adjoint is: $$ |A \cdot (\text{adj}(A))| = |A|^n $$
   This indicates that the determinant scales with the order of the matrix.

5. Property 4: Adjoint of the Adjoint
   The adjoint of the adjoint of ( A ) can be expressed as: $$ \text{adj}(\text{adj}(A)) = |A|^{n-2} A $$
   This shows a relation between the adjoint of an adjoint and the original matrix.

6. Property 5: Determinant of Adjoint of Adjoint
   For the adjoint of adjoint, we have: $$ |\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2} $$

7. Property 6: Adjoint of Product of Matrices
   For matrices ( A ) and ( B ): $$ \text{adj}(AB) = \text{adj}(B) \cdot \text{adj}(A) $$
   This property explains how to compute the adjoint of the product.

8. Property 7: Adjoint of Transpose
   The relationship between the adjoint and the transpose of a matrix is given by: $$ \text{adj}(A^T) = (\text{adj}(A))^T $$
   This shows that the adjoint of the transpose is the transpose of the adjoint.

9. Property 8: Adjoint with Scalar Multiplication
   If a scalar ( k ) multiplies the matrix ( A ): $$ \text{adj}(kA) = k^{n-1} \cdot \text{adj}(A) $$
   This property illustrates how scaling a matrix affects its adjoint.