# Find the determinant using row operations

## Homework Statement

find the determinant using row operations:
1 -2 2
0 5 -1
2 -4 1

## The Attempt at a Solution

i took row 3 and took 2 x row 1 away from it to get :
1 -2 2
0 5 -1
0 0 -3
1 x 5 x (-3) = -15...but i multiplied a row by 2 so i should get -30 for the det right?but the answer in my book is -15..what am i doing wrong?

Calculating the determinant for A, a 3x3 matrix with elements:
a b c
d e f
g h i

Det(A) = a(ei - fh) - b(di - fg) + c(dh-eg), by starting with row 1.

So -15 is the answer you should be getting.

Mark44
Mentor

Adding a row or a multiple of a row to another row doesn't change the value of the determinant. If you had replaced a row by a multiple of itself, then the determinant's value would have changed.

hmmm well i looked over my book, and it said if i multiply a row by k...i multiply the determinant by k...so when do we multiply the determinant by k?..cause what i did was that not multiplying a row by k(2)?

and thanks for ur help guys

Mark44
Mentor

hmmm well i looked over my book, and it said if i multiply a row by k...i multiply the determinant by k...so when do we multiply the determinant by k?..cause what i did was that not multiplying a row by k(2)?

and thanks for ur help guys
You did not replace a row by some multiple of itself; you added a multiple of a row to another row. These are different operations. On the other hand, if you had replaced row 1 by -2 times itself, and then added the first row to the third row, then your determinant would have been +30. This is not what you did though, since the first row stayed the same from start to finish.

In one of the linear algebra books I have, there is a theorem about determinants and row operations. The theorem has three parts.
1. If you interchange two rows, the determinant of the new matrix will be -1 times the determinant of the old matrix.
2. If you replace a row by k times itself, the determinant of the new matrix will be k times the determinant of the old matrix.
3. If you add k times one row to another row, the determinant of the new matrix will be equal to the determinant of the old matrix.