**Toolbox:**

- Let $ A = [ a_{ij} ] $ be a square matrix x. Let $ A_{ij}$ be the cofactor of $ a_{ij}$. Then $ [ A_{ij}]$ is the matrix of cofactors and $ adj\: A $ ( or adjoint of the matrix A) is given by $ adj\: A=[A_{ij}]^T$
- A determinant can be expanded by using the elements of any row or column.
- The inverse of a non-singular square matrix A is given by $ A^{-1} = \frac{1}{|A|} adj\: A.$ A non-singular matrix is one whose determinant value is nonzero.

Step 1

$ A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix} $

$ |A| = \begin{vmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{vmatrix} = 2(6-2)-2(2-1)+1(2-3)$

$ = 8-2-1=5 \neq 0$

$ \therefore A^{-1}$ exists.

Step 2

To find $ adj\: A$

$ [ A_{ij}] = \begin{bmatrix} (6-2) & -( 2-1) & (2-3) \\ -(4-2) & (4-1) & -(4-2) \\ (2-3) & -(2-1) & (6-2) \end{bmatrix} = \begin{bmatrix} 4 & -1 & -1 \\ -2 & 3 & -2 \\ -1 & -1 & 4 \end{bmatrix} $

$ adj\: A = [A_{ij} ]^T = \begin{bmatrix} 4 & -2 & -1 \\ -1 & 3 & -1 \\ -1 & 2 & -4 \end{bmatrix} $

Step 3

$ A^{-1} = \large\frac{1}{|A|} adj\: A = \large\frac{1}{5} \begin{bmatrix} 4 & -2 & -1 \\ -1 & 3 & -1 \\ -1 & 2 & -4 \end{bmatrix} $