# Without expanding, prove that = 0.

1
by Abhinanda
it is a determinant.

2015-04-23T02:05:42+05:30

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As we cannot evaluate the determinant by expansion, we have to reduce the determinant by row and column operations so that we can see that the determinant is zero.  Two columns or rows will be identical.

The operations  Column 1 - column2  and column 2 - column 3  are done.

25    21      219
15    27     198
21    17      181

row 1 - row 2      and  row 2  - row 3 :

10    -6     21
-6    10      17
21    17    181

row3 - 2 * row 1,      row2 + row1 and divide by 2

10    -6      21
2     2       19
1      29    139

row 3  - 4 * row 2

10    -6      21
2      2      19
-7     21     63

row 3  / 7    and  row1 + row 2,  then divide row 1  by 4.

3     -1      10
2      2      19
-1      3       9

Row 1  + row 3:
2    2      19
2    2      19
-1    3        9

Since two rows are equal, the determinant is 0.