Let A= \[\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\]
Then \[A^{-1}=\frac{adjA}{|A|}\]
Note -For inverse to exist matrix must be non singular . That means determinant of A must not be 0.
Adjoint of a matrix is defined as the transpose of cofactor matrix.
Calculation of minor of matrix
Step 1-It is calculated by deleting the all the elements of row and column corresponding to the element for which minor is determined.
step 2 -then find out the determinant of determinant of remaining sub matrix .The value of determinant is minor of that element.
Step 3-Step 1 and step 2 is repeated for all the element to get a minor of matrix of order same as that of order of given matrix
For example
Minor of a matrix A= \[\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\] is given by \[ \begin{bmatrix}a_{22}&a_{21}\\a_{12}&a_{11}\end{bmatrix}\]
Cofactor is \[(-1)^{i+j}\] where i represents the \[i^{th}\] row and \[j^{th}\] column.
Cofactor of A =\[ \begin{bmatrix}a_{22}& -a_{21}\\-a_{12}&a_{11}\end{bmatrix}\]
Transpose is defined as matrix obtained by changing corresponding rows with columns.If order of matrix A is \[ 2\times 3\] the the order of transpose of matrix will be \[ 3\times 2\]
Adjoint of a matrix is defined as the transpose of cofactor matrix.
transpose of cofactor matrix =\[ \begin{bmatrix}a_{22}& -a_{12}\\-a_{21}&a_{11}\end{bmatrix}\]
\[A^{-1}=\frac{adjA}{|A|}\]
