# Find the inverse of the following matrix : $\begin{bmatrix} 1 & 2 & -2 \\-1 & 3 & 0 \\0 & -2 & 1 \end{bmatrix}$

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
$A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$
$|A| = \begin{vmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{vmatrix} = 1(3-0)+1(2-4)+0$
$= 3-2 = 1 \neq 0$
$A$ is non-singular $\therefore A^{-1}$ exists.
Step 2
To find $adj\: A$
$[A_{ij}] = \begin{bmatrix} (3-0) & -(-1-0) & (2-0) \\ -(2-4) & (1-0) & -(-2-0) \\ (0+6) & -(0-2) & (3+2) \end{bmatrix} = \begin{bmatrix} 3 & 1 & 2 \\ 2 & 1 & 2 \\ 6 & 2 & 5 \end{bmatrix}$
$adj\: A = [A_{ij}]^T = \begin{bmatrix} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{bmatrix}$
Step 3
$A^{-1} = \large\frac{1}{|A|} adj\: A = \begin{bmatrix} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{bmatrix}$

answered May 21, 2013
edited May 21, 2013