How to find the inverse matrix of elementary matrix? Important process. . By the Grace of God

1, elementary matrix, inverse matrix or itself of row exchange (column exchange);

2. A row (or column) is multiplied by the multiple of the elementary matrix, and the inverse matrix is the multiple of the row (or column) divided by the elementary matrix;

3. A row (or column) is multiplied by a multiple and added to the elementary matrix of another row (or column). The inverse matrix is the inverse matrix of this row (or column) multiplied by this multiple, plus the elementary matrix of another row (or column).

The inverse matrix of elementary matrix is actually the same type of elementary matrix (which can be regarded as inverse transformation). For example, swap the positions of two rows (columns) in the matrix; Multiply a row (column) of the matrix by a non-zero constant k; Multiply one row (column) of a matrix by the constant k and add it to another row (column).

Extended data:

Elementary row transformation does not affect the solution of linear equations, and can also be used in Gaussian elimination to gradually transform the coefficient matrix into standard form. Elementary row transformation does not change the core of the matrix (so it does not change the solution set), but it changes the image of the matrix. On the contrary, elementary column transformation does not change the image, but changes the core.

Sometimes, when the order of a matrix is relatively high, using the value of its determinant and adjoint matrix to solve its inverse matrix will produce a lot of calculation. At this time, the original matrix with the same number of rows (also equal to the number of columns) is usually used side by side with identity matrix, and then the left side of this parallel matrix is transformed into identity matrix by elementary transformation. At this point, the matrix on the right is the inverse of the original matrix.

Baidu Encyclopedia-Elementary Matrix