Example of Finding the Inverse of a 2x2 Matrix
This solution has been done by the calculator presented on the site.
It is necessary to calculate a matrix A^{1}, inverse to the given one:
Formula for calculating the inverse matrix:
A^{1} = 1 / det A * 
 A_{11}  A_{21}   
A_{12}  A_{22} 
A_{11} ... A_{22} are numbers (algebraic additions) that will be calculated later.
It is impossible to divide by zero. Therefore, if the determinant of A is zero, then it is impossible to calculate inverse matrix.
Let's calculate the determinant A.
= 1 * 3  1 * ( 1) = 3 + 1 = 4
Determinant A is not zero. It is possible to calculate inverse matrix.
Let's calculate numbers (algebraic additions) A_{11} ... A_{22}



Row number 1 Column number 1 

Row 1 and column 1 have been deleted 

A_{11} 
= 
( 1) ^{1 + 1} 
* 
3

= 3 



Row number 1 Column number 2 

Row 1 and column 2 have been deleted 

A_{12} 
= 
( 1) ^{1 + 2} 
* 
1

= 1 



Row number 2 Column number 1 

Row 2 and column 1 have been deleted 

A_{21} 
= 
( 1) ^{2 + 1} 
* 
1

= 1 



Row number 2 Column number 2 

Row 2 and column 2 have been deleted 

A_{22} 
= 
( 1) ^{2 + 2} 
* 
1

= 1 
Result:
A^{1} = 1 / det A * 
 A_{11}  A_{21}   
A_{12}  A_{22} 
A^{1} = 1 / 4 * 
 3  1   
1  1 
A^{1} = 
 3/4  1/4   
1/4  1/4 
It is necessary to check that A^{1} * A = E.
We will use the penultimate form of the inverse matrix A^{1}. This will allow us to count without fractions.
 *   = 
 b_{11}  b_{12}   b_{21}  b_{22}  
b_{11} = 3 * 1 + ( 1) * ( 1) =
3 + 1 = 4
b_{12} = 3 * 1 + ( 1) * 3 =
3  3 = 0
b_{21} = 1 * 1 + 1 * ( 1) =
1  1 = 0
b_{22} = 1 * 1 + 1 * 3 =
1 + 3 = 4
It is necessary to multiply the result by 1/4
Thus, the found matrix A^{1} is inverse for the given matrix A.