Question

Use elementary row or column operations to evaluate the following determinant. You may use a calculator to do the multiplications. 1 -9 6 -9 -5 2 -17 6 -21 -11 2 -13 4 -17 -12 0 6 4 2 0 0 -2 16 8 2

Answer 1

Here, 25 numbers have been listed. We presume that the det(A) is required, where A is a 5x5 matrix , which is as under:

1	-9	6	-9	-5
2	-17	6	-21	-11
2	-13	4	-17	-12
0	6	4	2	0
0	-2	16	8	2

Let us apply the following row operations to A.

1.Add -2 times the 1st row to the 2nd row

2.Add -2 times the 1st row to the 3rd row

3.Add -5 times the 2nd row to the 3rd row

4.Add -6 times the 2nd row to the 4th row

5.Add 2 times the 2nd row to the 5th row

6.Multiply the 3rd row by 1/22

7.Add -40 times the 3rd row to the 4th row

8.Add -4 times the 3rd row to the 5th row

9.Multiply the 4th row by -11/100

10.Add 10/11 times the 4th row to the 5th row

11. Multiply the 5th row by -5/3

Then A changes to M =

1	-9	6	-9	-5
0	1	-6	-3	-1
0	0	1	8/11	3/22
0	0	0	1	-3/50
0	0	0	0	1

It may be observed that M is an upper triangular matrix.

We know that:

1. The interchanging of two rows or columns of a determinant changes the sign of the determinant.
2. Adding a multiple of one row to another does not have any effect on the value of the determinant.
3. Multiplying a row of a determinant by a constant, scales up the value of the determinant by that constant.
4. The determinant of a lower triangular matrix/ an upper triangular matrix is the product of the diagonal entries.

Thus, det(M) = 1. It may also be observed that only the 6^th,9^th and 11^th row opearations affect the value of det(A), and det(M) = (1/22)(-11/100)(-5/3) det(A)= (1/120) det(A) so that det(A) = 120det(M) = 120.