For homogeneous equations, the form is AX=0.

When r(A)=n, that is |A|! =0, the equation has only 0 solution. ("! = "Not equal to)

When r (a) < N, the number of solutions =n-r(A).

For nonhomogeneous linear equations, the form is ax = b.

The augmented matrix is introduced. A'=Ab is called the augmented matrix of A, where Ab is not a multiplication, but a matrix obtained by placing the column vector B on the right side of matrix A. ..

When r(A')=r(A)=n, the equation has a unique solution.

When r (a') = r (a)

When r(A')! =r(A), the equation has no solution.

Secondly, it is the process of solving.

It is transformed into a trapezoidal matrix by row-column transformation, and then solved.

Row-column conversion is to multiply the n of one row by another. N is a real number.

Shape image of trapezoidal matrix

1 2 3

0 5 6

0 0 3

or

1 2 3

0 5 6

0 0 0

The general method is to multiply the n in the first row by the second and third rows, so that the first number in the second and third rows becomes 0. Then multiply the n in the second row by the third row, so that the second number in the third row becomes 0. And so on.

I finally understand.

| 1 2 3| x 1

|0 5 6| x2 = 0 or b (0 and b are column vectors, that is, they are not a number).

|0 0 3| x3

This equation can be solved. The solutions of this homogeneous equation are x3= 1, x2=-6/5 and x3=3/5.