It is known that the elements in the first row of the fourth-order determinant D are 1,2,0, - 4 respectively, and the remainder of the third row is 6, x, 19, 2 in turn. Try to find the root of X

It is known that the elements in the first row of the fourth-order determinant D are 1,2,0, - 4 respectively, and the remainder of the third row is 6, x, 19, 2 in turn. Try to find the root of X

Related corollaries of determinant expansion theorem: the sum of the elements in one row multiplied by the algebraic cofactors of the elements in another row is equal to 0
So there are:
1*6 + 2X + 0*19 -4*2 = 0
The solution is x = 1

Determinant d = | 1,2,3,4 | 2,3,4,1 | 3,4,1,2 | 4,1,2,3|
Calculate the following determinant
First line 1,2,3,4
Second line 2,3,4,1
The third line is 3,4,1,2
The fourth line is 4,1,2,3

Step 1: add columns 2, 3 and 4 to column 1, propose the common factor 10 of column 1, and change it into
1 2 3 4
1 3 4 1
1 4 1 2
1 1 2 3
Step 2: multiply the first line by - 1 and add it to the other lines
1 2 3 4
0 1 1 -3
0 2 -2 -2
0 -1 -1 -1
Step 3: R3 - 2r1, R4 + R1
1 2 3 4
0 1 1 -3
0 0 -4 4
0 0 0 -4
So determinant = 10 * (- 4) * (- 4) = 160
Please accept if you are satisfied^_^

Integers smaller than 1 are arranged as follows: first column, second column, third column, fourth column
Integers smaller than 1 are arranged as follows:
First column second column third column fourth column
－2 －3 －4 －5
－9 －8 －7 －6
－10 －11 －12 －13
－17 －16 －15 －14
............
In the above numbers, observe their rules and answer which column is the number - 100?

The fourth column, 100 / 4 equal to 25, belongs to 2,3,4,5. Just divide, so it's the fourth column, give points!