How to transform the fourth-order determinant into upper triangle or lower triangle?

How to transform the fourth-order determinant into upper triangle or lower triangle?

Note the properties of the determinant: 1. Row and column exchange, the value of the determinant remains unchanged; 2. Exchange the position of any two rows or columns, the inverse sign of the determinant; 3. Add one row of the determinant multiplied by a constant to another row or one column multiplied by a constant to another column

How does this determinant become an upper triangular determinant?
1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3

The elementary change of a row is to eliminate the number under the diagonal line. For example, the first row is multiplied by - 2 and added to the second row. In this way, the 2 of the second row can be deleted. Similarly, the first number of the third and fourth rows can be deleted. Then, starting from the second row, the second number of the third and fourth rows can be deleted in the same way, and so on. Finally, the third number of the fourth row can be deleted, which becomes the upper triangular determinant

Given that the determinant of second order matrix A is negative, it is proved that a can be similar to diagonal matrix

The conclusion is only true for real matrix, when the two eigenvalues are not equal

Find the trajectory equation of a point whose distance is equal to that of a (1,2), B (- 3,4)
Can you use the method of finding the perpendicular of AB

Suppose P (x, y)
PA²=PB²
So (x-1) & sup2; + (Y-2) & sup2; = (x + 3) & sup2; + (y-4) & sup2;
2x-y+5=0