# Notes - Rings and Modules HT24, Matrices over a ring

### Flashcards

Suppose $A$ is a matrix over a ring $R$. How is the adjugate matrix $\text{adj}(A)$ defined, and what useful relation is there involving the determinant?

where $M$ is the matrix of minors. We have that

\[A \text{adj}(A) = \det(A) \cdot I\]Suppose $A$ is a matrix over a ring $R$. When is $A$ invertible?

When $\det A$ is a unit.

Quickly prove that if $A$ is a matrix over a ring $R$, then it is invertible if and only if $\det A$ is a unit.

Suppose $A$ is invertible with inverse $B$, so that $AB = I$. Then

\[\det(AB) = \det(A) \det(B) = \det(I) = 1\]Hence $\det A$ and $\det B$ must both be units.

Now suppose that $\det A$ is a unit. Then invertibility follows from the identity

\[A \cdot \text{adj}(A) = \det(A) \cdot I\]Recall that

\[\text{adj}(A) _ {ij} = (-1)^{i+j} M _ {ji}\]where $M$ is the matrix of minors. To prove the identity, let $B = A \cdot \text{adj}(A)$. Then

\[B _ {ij} = \sum^n _ {k = 1} a _ {ik} (-1)^{j + k} M _ {jk}\]When $i = j$, this correspondings to the Laplace expansions along row $i$, so $B _ {ii} = \det A$ for all $i$.

When $i \ne j$, we have

\[\begin{aligned} B_ {ij} &= \sum^n_ {k = 1} a_ {ik} (-1)^{j + k} M_ {jk} \\\\ &= \sum^n_ {k = 1} a'_ {jk} (-1)^{j + k} M'_ {jk} \\\\ &= 0 \end{aligned}\]where $a’ _ {jk}$ denotes the elements of $A’ _ i$, which is $A$ but where row $i$ has been replaced by row $j$, and likewise $M’ _ {jk}$ is defined similarly. Then $\det(A’ _ i) = 0$ since it has a duplicate row. Then

\[(A \cdot \text{adj}(A))_ {ij} = B_ {ij} = \begin{cases} 0 &\text{if } i \ne j \\\\ \det A &\text{if } i = j \end{cases}\]so

\[A \cdot \text{adj}(A) = \det(A) \cdot I\]as required.

If $A, B \in \text{Mat} _ {n, m}(R)$, what does it mean for them to be equivalent, and how is this related to another notion of equivalence for matrices?

- $A \sim B$ iff $B = PAQ$ where $P \in \text{Mat} _ {n, n}(R)$ and $Q \in \text{Mat} _ {m, m}(R)$ are invertible
- This is the same notion as two matrices being ERC equivalent, where there is a sequence of elementary row and column operations that transforms one into another