The given determinant
Explanation. In the first step of the reduction keep the second column unaltered; for the first new column diminish each constituent of the first column by the corresponding constituent of the second; for the third new column diminish each constituent of the third column by the corresponding constituent of the second. In the second step take out the factors 3 and -4. In the third step keep the first row unaltered; for the second new row diminish the constituents of the second by the corresponding ones of the first; for the third new row diminish the constituents of the third by twice the corresponding constituents of the first. The remaining steps will be easily seen.
Ex. 2. Show that | a-b-c 2a 2a = (a+b+c)^3. 2b b-c-a 2b 2c 2c c-a-b
The given determinant
Explanation. In the first new determinant the first row is the sum of the constituents of the three rows of the original determinant, the second and third rows being unaltered. In the third of the new determinants the first column remains unaltered, while the second and third columns are obtained by subtracting the constituents of the first column from those of the second and third respectively. The remaining transformations are sufficiently obvious.
