Reduce the given determinant to upper triangular form and th

Reduce the given determinant to upper triangular form and them evaluate. det(1 0 -1 3 2 2 0 0 1 0 4 -1 0 1 -5 1)

Solution

Let A =

1

0

-1

3

2

2

0

0

1

0

4

-1

0

1

-5

1

Let us perform the following row operations on A:

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

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

3. Multiply the 2nd row by ½

3. Add -1 times the 2nd row to the 4th row

4. Multiply the 3rd row by 1/5

5. Add 6 times the 3rd row to the 4th row

Then A changes to B =

1

0

-1

3

0

1

1

-3

0

0

1

-4/5

0

0

0

-4/5

We know that the determinant of an upper triangular matrix is the product of the diagonal elements. Hence det(B) = -4/5.

We also know that:

Hence det(B) = (1/2)(1/5)det(A) = 1/10 det(A). Therefore, det(A) = 10*det(B) = 10*(-4/5) = -8.

1	0	-1	3
2	2	0	0
1	0	4	-1
0	1	-5	1