Inverse of a Matrix by the Adjoint Method

Finding the inverse of a matrix is one of the most practically useful skills in this CHSE Odisha Business Mathematics and Statistics chapter — it is what makes solving a system of linear equations by matrix methods possible (next section), and it depends on ideas (determinant, cofactor) that are worth recapping briefly here even though they are treated fully in the Determinants chapter.

Determinant and singularity — a quick recap

For a $2\times2$ matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, its determinant is $|A|=ad-bc$. A square matrix $A$ is called non-singular if $|A|\neq0$, and singular if $|A|=0$. Only a non-singular matrix has an inverse — this is the single most important fact in this section.

Minors, cofactors and the adjoint

For a $3\times3$ (or larger) matrix, the minor $M_{ij}$ of element $a_{ij}$ is the determinant left after deleting row $i$ and column $j$; the cofactor is $C_{ij}=(-1)^{i+j}M_{ij}$ (a sign that alternates in a checkerboard pattern). The adjoint of $A$, written $\text{adj}(A)$, is the TRANSPOSE of the matrix of cofactors — cofactor $C_{ij}$ goes into position $(j,i)$, not $(i,j)$.

The inverse formula

For any non-singular square matrix $A$,

Inverse of a 2×2 matrix

For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ with $|A|=ad-bc\neq0$, the adjoint is found by a simple shortcut — swap the diagonal elements and change the sign of the off-diagonal elements:

Inverse of a 3×3 matrix

For a $3\times3$ matrix, all nine cofactors $C_{11},C_{12},\ldots,C_{33}$ must be computed individually (each from a $2\times2$ minor), arranged into the cofactor matrix, then transposed to give $\text{adj}(A)$, and finally divided by $|A|$. This is more work than the $2\times2$ shortcut but follows exactly the same formula. …