Example of Finding the Inverse of a 3x3 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_{31}   
A_{12}  A_{22}  A_{32} 
A_{13}  A_{23}  A_{33} 
A_{11} ... A_{33} 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.
det A = 
 4  3  2   = 
2  1  1 
3  3  2 
The elements of row 3 multiplied by 1 are added to the corresponding elements of row 1.
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 4 + 3 * ( 1)  3 + 3 * ( 1)  2 + 2 * ( 1)  
2  1  1 
3  3  2 
This elementary transformation does not change the value of the determinant.
Expand the determinant along the row 1.
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Row number 1 Column number 1 

Element 

Row 1 and column 1 have been deleted 
( 1) ^{1 + 1} 
* 
1 
* 


Row number 1 Column number 2 

Element 

Row 1 and column 2 have been deleted 
( 1) ^{1 + 2} 
* 
0 
* 


Row number 1 Column number 3 

Element 

Row 1 and column 3 have been deleted 
( 1) ^{1 + 3} 
* 
0 
* 

Products are summed. If the element is zero than product is zero too.
= ( 1) ^{1 + 1} * 1 * 
 1  1   = 
3  2 
= 5
Determinant A is not zero. It is possible to calculate inverse matrix.
Let's calculate numbers (algebraic additions) A_{11} ... A_{33}



Row number 1 Column number 1 

Row 1 and column 1 have been deleted 
A_{11} 
= 
( 1) ^{1 + 1} 
* 

= 1 * 2  ( 1) * 3 = 2 + 3 = 5



Row number 1 Column number 2 

Row 1 and column 2 have been deleted 
A_{12} 
= 
( 1) ^{1 + 2} 
* 

=  ( 2 * 2  ( 1) * 3 ) =  (4 + 3) = 7



Row number 1 Column number 3 

Row 1 and column 3 have been deleted 
A_{13} 
= 
( 1) ^{1 + 3} 
* 

= 2 * 3  1 * 3 = 6  3 = 3



Row number 2 Column number 1 

Row 2 and column 1 have been deleted 
A_{21} 
= 
( 1) ^{2 + 1} 
* 

=  ( 3 * 2  2 * 3 ) =  (6  6) = 0



Row number 2 Column number 2 

Row 2 and column 2 have been deleted 
A_{22} 
= 
( 1) ^{2 + 2} 
* 

= 4 * 2  2 * 3 = 8  6 = 2



Row number 2 Column number 3 

Row 2 and column 3 have been deleted 
A_{23} 
= 
( 1) ^{2 + 3} 
* 

=  ( 4 * 3  3 * 3 ) =  (12  9) = 3



Row number 3 Column number 1 

Row 3 and column 1 have been deleted 
A_{31} 
= 
( 1) ^{3 + 1} 
* 

= 3 * ( 1)  2 * 1 = 3  2 = 5



Row number 3 Column number 2 

Row 3 and column 2 have been deleted 
A_{32} 
= 
( 1) ^{3 + 2} 
* 

=  ( 4 * ( 1)  2 * 2 ) =  (4  4) = 8



Row number 3 Column number 3 

Row 3 and column 3 have been deleted 
A_{33} 
= 
( 1) ^{3 + 3} 
* 

= 4 * 1  3 * 2 = 4  6 = 2
Result:
A^{1} = 1 / det A * 
 A_{11}  A_{21}  A_{31}   
A_{12}  A_{22}  A_{32} 
A_{13}  A_{23}  A_{33} 
A^{1} = 1 / 5 * 
 5  0  5   
7  2  8 
3  3  2 
A^{1} = 
 1  0  1   
7/5  2/5  8/5 
3/5  3/5  2/5 
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_{13}   b_{21}  b_{22}  b_{23}  b_{31}  b_{32}  b_{33}  
b_{11} = 5 * 4 + 0 * 2 + ( 5) * 3 =
20 + 0  15 = 5
 *   = 
 5  b_{12}  b_{13}   b_{21}  b_{22}  b_{23}  b_{31}  b_{32}  b_{33}  
b_{12} = 5 * 3 + 0 * 1 + ( 5) * 3 =
15 + 0  15 = 0
 *   = 
 5  0  b_{13}   b_{21}  b_{22}  b_{23}  b_{31}  b_{32}  b_{33}  
b_{13} = 5 * 2 + 0 * ( 1) + ( 5) * 2 =
10 + 0  10 = 0
 *   = 
 5  0  0   b_{21}  b_{22}  b_{23}  b_{31}  b_{32}  b_{33}  
b_{21} = 7 * 4 + 2 * 2 + 8 * 3 =
28 + 4 + 24 = 0
 *   = 
 5  0  0   0  b_{22}  b_{23}  b_{31}  b_{32}  b_{33}  
b_{22} = 7 * 3 + 2 * 1 + 8 * 3 =
21 + 2 + 24 = 5
 *   = 
 5  0  0   0  5  b_{23}  b_{31}  b_{32}  b_{33}  
b_{23} = 7 * 2 + 2 * ( 1) + 8 * 2 =
14  2 + 16 = 0
 *   = 
 5  0  0   0  5  0  b_{31}  b_{32}  b_{33}  
b_{31} = 3 * 4 + ( 3) * 2 + ( 2) * 3 =
12  6  6 = 0
 *   = 
 5  0  0   0  5  0  0  b_{32}  b_{33}  
b_{32} = 3 * 3 + ( 3) * 1 + ( 2) * 3 =
9  3  6 = 0
b_{33} = 3 * 2 + ( 3) * ( 1) + ( 2) * 2 =
6 + 3  4 = 5
It is necessary to multiply the result by 1/5
Thus, the found matrix A^{1} is inverse for the given matrix A.