[[Linear algebra MOC]]
# Adjugate matrix

The **adjugate matrix** $\opn{adj}T$ of a [[Matrix algebra over a ring|square matrix]] $T \in \opn M_{n , n}(R)$ is the [[Matrix transpose|transpose]] of its [[Minor|cofactor matrix]] $\opn{cof}T$. #m/def/linalg 

## Properties

For $T \in M_{n,n}(R)$, we have

1. $(\opn{adj} T) T = T(\opn{adj}T) = (\det T) 1_{n}$ by the [[Laplace expansion of the matrix determinant]]

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