Finding Inverse Matrix by Elementary Row Transformation

That is, the matrix (a, e) is transformed into the form of (e, b) by row transformation, and then b is equal to the inverse of a.

here

(A，E)= 1

0 1 2 1 0 0

1 1 - 1 0 1 0

2 4 0 0 0 1 r3-2r2，r2-r 1

~

0 1 2 1 0 0

1 0 -3 - 1 1 0

0 2 2 0 -2 1 r3-r 1，r 1-r3

~

0 0 2 2 2 - 1

1 0 -3 - 1 1 0

010-1-21r1/2, R2+3r 1, exchange sort.

~

1 0 0 2 4 -3/2

0 1 0 - 1 -2 1

0 0 1 1 1 - 1/2

In this way, (a, e) ~ (e, a- 1) is transformed by elementary lines.

So we get that the inverse of the original matrix is

2 4 -3/2

- 1 -2 1

1 1 - 1/2

Extended data:

(1) exchange two rows of the matrix (I, J exchange, two rows are denoted as ri and RJ);

(2) Multiplying all the elements in a row of the matrix by a non-zero number k (the multiplication of the i-th row by k means ri× k);

(3) Multiply all the elements in one row of the matrix by a number k and add it to the corresponding elements in another row (the j-th row is multiplied by k and added to the i-th row with the mark ri+krj).

Similarly, the above "row" is changed to "column" to get the definition of matrix elementary transformation, and the corresponding symbol "R" is changed to "C".

Elementary row transformation and elementary column transformation of matrix are collectively called elementary transformation of matrix.

Baidu Encyclopedia-Matrix Transformation