Summary

Adjoint. $\operatorname{adj}A=$ transpose of the cofactor matrix of $A$.

Central identity. $A(\operatorname{adj}A)=(\operatorname{adj}A)A=|A|\,I_n$.

Inverse formula. $A^{-1}=\dfrac1{|A|}\operatorname{adj}A$.

Standing laws (non-singular $A,B$ of the same order, $\lambda\ne0$): (i) $|A^{-1}|=\dfrac1{|A|}$; (ii) $(A^T)^{-1}=(A^{-1})^T$; (iii) $(\lambda A)^{-1}=\dfrac1\lambda A^{-1}$; (iv) $(AB)^{-1}=B^{-1}A^{-1}$; (v) $(A^{-1})^{-1}=A$.

Adjoint identities (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) $(\operatorname{adj}A)^T=\operatorname{adj}(A^T)$; (vii) $\operatorname{adj}(AB)=(\operatorname{adj}B)(\operatorname{adj}A)$.

Orthogonal matrix. $A$ orthogonal $\iff AA^T=A^TA=I\iff A$ non-singular and $A^{-1}=A^T$.

Methods for $AX=B$ ($A$ square, non-singular unless noted).

1. Matrix inversion: $X=A^{-1}B$.
2. Cramer's rule: $x=\Delta_1/\Delta,\ y=\Delta_2/\Delta,\ z=\Delta_3/\Delta$ ($\Delta\ne0$).
3. Gaussian elimination: row-reduce $[A|B]$ to echelon form, back-substitute (works even if $A$ is singular or non-square). …