Example ¹2. Finding the Inverse of a 3x3 MatrixThis solution was made using 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_{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.
The elements of row 3 multiplied by 2 are added to the corresponding elements of row 2. more info
This elementary transformation does not change the value of the determinant.
Expand the determinant along the row 2. more info
Products are summed. If the element is zero than product is zero too.
=  5 * ( 1 * ( 1)  3 * ( 1) ) = =  5 * ( 1 + 3 ) = = 10 Determinant A is not zero. It is possible to calculate inverse matrix. Let's calculate numbers (algebraic additions) A_{11} ... A_{33}
= 2 * 2  1 * ( 1) = 4 + 1 = 5
=  ( 2 * 2  1 * ( 1) ) =  (4 + 1) = 5
= 2 * ( 1)  2 * ( 1) = 2 + 2 = 0
=  ( 3 * 2  1 * ( 1) ) =  (6 + 1) = 7
= 1 * 2  1 * ( 1) = 2 + 1 = 3
=  ( 1 * ( 1)  3 * ( 1) ) =  (1 + 3) = 2
= 3 * 1  1 * 2 = 3  2 = 1
=  ( 1 * 1  1 * 2 ) =  (1  2) = 1
= 1 * 2  3 * 2 = 2  6 = 4 Result:
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} = 5 * 1 + ( 7) * 2 + 1 * ( 1) =
5  14  1 = 10
b_{12} = 5 * 3 + ( 7) * 2 + 1 * ( 1) =
15  14  1 = 0
b_{13} = 5 * 1 + ( 7) * 1 + 1 * 2 =
5  7 + 2 = 0
b_{21} = 5 * 1 + 3 * 2 + 1 * ( 1) =
5 + 6  1 = 0
b_{22} = 5 * 3 + 3 * 2 + 1 * ( 1) =
15 + 6  1 = 10
b_{23} = 5 * 1 + 3 * 1 + 1 * 2 =
5 + 3 + 2 = 0
b_{31} = 0 * 1 + ( 2) * 2 + ( 4) * ( 1) =
0  4 + 4 = 0
b_{32} = 0 * 3 + ( 2) * 2 + ( 4) * ( 1) =
0  4 + 4 = 0
b_{33} = 0 * 1 + ( 2) * 1 + ( 4) * 2 =
0  2  8 = 10
It is necessary to multiply the result by 1/10
Thus, the found matrix A^{1} is inverse for the given matrix A. Please do not forget to support the site with a link. © 20102023 If you have any comments, please write to siteReshmat@yandex.ru 
