Properties of Inverses of Matrices

Collecting the working laws obeyed by inverses (for non-singular square matrices $A,B$ of the same order, and $\lambda\ne0$ a scalar):

Theorem 1.4. (i) $|A^{-1}|=\dfrac1{|A|}$. (ii) $(A^T)^{-1}=(A^{-1})^T$. (iii) $(\lambda A)^{-1}=\dfrac1\lambda A^{-1}$.

Proof sketch of (i). From $AA^{-1}=I_n$, the product rule gives $|A||A^{-1}|=1$, so $|A^{-1}|=1/|A|$.

Theorem 1.5 (left cancellation). If $A$ is non-singular and $AB=AC$, then $B=C$ (pre-multiply both sides by $A^{-1}$).

Theorem 1.6 (right cancellation). If $A$ is non-singular and $BA=CA$, then $B=C$ (post-multiply both sides by $A^{-1}$).

Theorem 1.7 (reversal law). If $A,B$ are non-singular of the same order, $AB$ is also non-singular and

(the order flips -- exactly parallel to the transpose law $(AB)^T=B^TA^T$). Proof idea: check directly that $(B^{-1}A^{-1})(AB)=B^{-1}(A^{-1}A)B=B^{-1}IB=B^{-1}B=I$, and similarly the other way.

Theorem 1.8 (double inverse). If $A$ is non-singular, so is $A^{-1}$, and $(A^{-1})^{-1}=A$.

Theorem 1.9 (adjoint identities). For non-singular $A$ of order $n$: (i) $\operatorname{adj}(A^{-1})=(\operatorname{adj}A)^{-1}=\dfrac{A}{|A|}$; (ii) $|\operatorname{adj}A|=|A|^{n-1}$; (iii) $\operatorname{adj}(\operatorname{adj}A)=|A|^{n-2}A$; (iv) $\operatorname{adj}(\lambda A)=\lambda^{n-1}\operatorname{adj}A$; (v) $|\operatorname{adj}(\operatorname{adj}A)|=|A|^{(n-1)^2}$; (vi) $(\operatorname{adj}A)^T=\operatorname{adj}(A^T)$.

Theorem 1.10. For non-singular $A,B$ of the same order, $\operatorname{adj}(AB)=(\operatorname{adj}B)(\operatorname{adj}A)$ -- order reverses, just like the inverse and the transpose.

Application to geometry. Rotating the coordinate axes through angle $\theta$ transforms $(x,y)$ to $(X,Y)$ via $\begin{pmatrix}x\\y\end{pmatrix}=W\begin{pmatrix}X\\Y\end{pmatrix}$ with $W=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}$. Since $W^TW=WW^T=I_2$ (using $\cos^2\theta+\sin^2\theta=1$), $W$ satisfies $W^{-1}=W^T$ -- a Definition 1.3: orthogonal matrix is exactly a square matrix $A$ with $AA^T=A^TA=I$, equivalently ($A$ non-singular and) $A^{-1}=A^T$. …