Solution: Consider the following complex system of third-order linear equations: 3x2y-z=72x-4y5z=-6-xy2z=10 We represent this system of equations in the form of an augmented matrix so that we can solve it using Gaussian elimination. The augmentation matrix is a splicing of the original coefficient matrix and a constant sequence, as follows: 32-1|7||2-45|-6||-112|10|Step 1: Simplification (elimination stage) 1, in order to solve the first unknown x, we need to convert the first.
The second and third rows are multiplied by 1 3 and 1, respectively, and the first row is subtracted so that the first column elements of the second and third rows become 0. Get: 32-1|7||0-88|-19||015|3|2. Then, in order to eliminate the second element of the third row, multiply the third row by -1 8 and add it to the second row, so that the second element of the second row becomes 0. ``32-1|7||0-80|-11||015|3|Step 2: Solve 1 by regression, we can first solve the value of z. Since the third line is already in a simpler form, it is possible to get z=3 5 directly. 2. Substituting z=3 5 into the second row of equations, we get y=(1183 5) (-8)=1. 3. Substituting y=1 and z=3 5 into the first row of equations, we get x=(7-2113 5) 3=2. The solution of this system of third-order linear equations is x=2, y=1, z=3 5.
Through the Gaussian elimination method, we gradually simplified the original linear equations, converted them into stepped matrices, and finally succeeded in finding the exact solutions of all unknowns by the method of regression. This method is suitable for any system of nth-order linear equations, where the solution can be found or no solution can be determined as long as the matrix is not in order.