Adjoint, Inverse and Matrix Polynomial

Adjoint, Inverse and Matrix Polynomial

This lesson covers the following key concepts:

• Adjoint (adjugate) of a matrix
• Singular and non-singular matrices
• Inverse of a matrix: A⁻¹ = adj(A)/|A|
• Properties of inverse
• Matrix polynomial
• Characteristic equation of a matrix
• Cayley-Hamilton theorem
• Finding A⁻¹ and Aⁿ using Cayley-Hamilton

Important Formulas

• $$\text{adj}(A) = [A_{ij}]^T$$
• $$A \cdot \text{adj}(A) = |A|I$$
• $$A^{-1} = \frac{\text{adj}(A)}{|A|} \text{ if } |A| \neq 0$$
• $$(AB)^{-1} = B^{-1}A^{-1}$$
• $$\text{Characteristic eq: } |A - \lambda I| = 0$$
• $$\text{Cayley-Hamilton: A satisfies its own characteristic eq}$$
• $$A^2 - (\text{tr}A)A + |A|I = 0 \text{ (for 2×2)}$$