Inverse of a Matrix — by Adjoint and by Elementary Transformations

For a square matrix $A$, an inverse is a matrix $A^{-1}$ (of the same order) such that $A A^{-1}=A^{-1}A=I,$ where $I$ is the unit matrix. A square matrix has an inverse if and only if its determinant is non-zero; such a matrix is called non-singular, and a matrix with $|A|=0$ is singular and has no inverse. When the inverse exists it is unique.

Method 1 — the adjoint method. The adjoint of $A$, written $\operatorname{adj}(A)$, is the transpose of the matrix of cofactors of $A$. Recall the cofactor of the entry in row $i$, column $j$ is $C_{ij}=(-1)^{i+j}M_{ij}$, where $M_{ij}$ is the corresponding minor. Then $A^{-1}=\frac{1}{|A|}\,\operatorname{adj}(A),\qquad |A|\neq 0.$

For a $2\times 2$ matrix there is a quick special case: if $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ then $|A|=ad-bc$ and $A^{-1}=\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ — swap the diagonal entries, change the sign of the other two, and divide by the determinant. For a $3\times 3$ matrix the nine cofactors are computed, arranged, and transposed to give the adjoint before dividing by $|A|$. …