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How to find the rank of matrix
In general, when the number of rows and columns are both high, it is very troublesome to find the rank according to the definition. For a row ladder matrix, its rank is obviously equal to the number of rows of non-zero rows. Because the rank of two equivalent matrices is equal, we can also use elementary transformation to matrix
Rank of matrix
2 4 1 0
1 0 3 2
-1 5 -3 1
0 1 0 2
r1-2r2,r3+r2
0 4 -5 -4
1 0 3 2
0 5 0 3
0 1 0 2
r1-4r4,r3-5r4
0 0 -5 4
1 0 3 2
0 0 0 -7
0 1 0 2
The rank is 4
To find the rank of a matrix is to use elementary row transformation to transform more rows into the form of 0 vector. Is there any idea or skill? Blindly, we can't get the answer
Firstly, non-zero elements are placed in the upper left corner of the matrix (a11) by row column exchange, and then - A12 / a11, - A13 / a11, - A14 / a11... Times of the first row are added to the corresponding column by using the elements in the upper left corner, then the elements in the first column can be changed to 0 except for the first. Then, A22 is not zero by row column exchange, and - A23 / A22, - A24 / A22, - a25 / A22... Times of the second row are added to the corresponding row, Then the second column except A12, A22 are all changed to 0. Then the row and column exchange makes A33 not 0, and the third row's - A34 / A33, - A35 / A33, - A36 / A33... Times are added to the corresponding column, and so on. This method is not the simplest, but it is omnipotent. Making this kind of transformation can certainly turn the matrix into row ladder type
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