Determinants - Adjoint and inverse of a square matrix

Minors and Cofactors: The minor $M_{ij}$ of an element $a_{ij}$ of a determinant is the determinant of the sub-matrix left after deleting its $i$-th row and $j$-th column. The cofactor $C_{ij}$ is defined as $(-1)^{i+j} M_{ij}$. Visually, you can imagine 'cross-out' lines passing through the element $a_{ij}$ horizontally and vertically; the remaining numbers form the minor.

Adjoint of a Square Matrix: The adjoint of a square matrix $A = [a_{ij}]$ is the transpose of the matrix of its cofactors. It is denoted by $adj A$. Visually, if you arrange all cofactors $C_{ij}$ into a matrix, the adjoint is obtained by 'reflecting' that matrix across its main diagonal (switching rows and columns).

Singular and Non-singular Matrices: A square matrix $A$ is called singular if its determinant $|A| = 0$. It is called non-singular if $|A| \neq 0$. Geometrically, a singular matrix collapses the area or volume it represents to zero, meaning it cannot be 'undone' or inverted.

Existence of Inverse: A square matrix $A$ has an inverse $A^{-1}$ if and only if $A$ is a non-singular matrix (i.e., $|A| \neq 0$). The inverse is the unique matrix such that $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

Product of Matrix and Adjoint: For any square matrix $A$ of order $n$, the product $A(adj A)$ or $(adj A)A$ results in a scalar matrix $|A|I$. Visually, this product yields a diagonal matrix where every entry on the main diagonal is the value of the determinant $|A|$, and all other entries are zero.

Properties of Adjoint Determinants: The determinant of the adjoint of a matrix $A$ of order $n$ is given by $|adj A| = |A|^{n-1}$. This means the 'scaling factor' of the adjoint is the determinant raised to a power one less than the matrix dimension.

Reversal Law for Inverses: For two non-singular matrices $A$ and $B$ of the same order, $(AB)^{-1} = B^{-1}A^{-1}$. This 'socks-and-shoes' principle implies that to undo a sequence of transformations, you must undo the last one first.