# Finding a determinant

1. Jan 29, 2010

### 60051

1. The problem statement, all variables and given/known data

Find the determinant of the following matrix:

4.....0.....1
19...1....-3
7.....1.....0

I chose the 1st row to do the operations on.

4 [(1*0) - 1*(-3)] + 1 [19*1 - 7*1]

= 4[0 - (-3)] + 1[12]
=12 + 12
=24

I can't see any mistakes in that, but it's apparently wrong. The answer is supposed to be 0. Here's the thing, for some rows/colums, the answer comes out to be 24, while for other rows/colums, the answer is 0. Shouldn't it not matter which row/column you choose? So why am I getting different answers?

For example, if I choose the 3rd column...

1 [19 - 7] - 3 [4 - 0]
= 12 - 12
= 0

So why am I getting different answers?

2. Jan 29, 2010

### Staff: Mentor

Are you familiar with the method of visualizing the diagonal multiplications? That's the way I prefer to do it, and it does give an answer of zero.

See the 3x3 matrix determinant example part-way down this page:

http://en.wikipedia.org/wiki/Determinant

.

3. Jan 29, 2010

### Staff: Mentor

I get a determinant of 24 in two ways: expanding the first row; expanding the 3rd column. There is a sign error in your work in expanding the third column.
It should be
1 [19 - 7] - (-3) [4 - 0]
= 12 + 12
= 24

4. Jan 29, 2010

### Staff: Mentor

Ack! I dropped that "-" sign as well. Thanks Mark.

5. Jan 29, 2010

### Staff: Mentor

Happens to us all... leastwise it happens to me!

6. Jan 29, 2010

### 60051

The answer given is 0, even though 24 also works, as we have seen.

So what's the deal? Are there two determinants?

7. Jan 29, 2010

### Staff: Mentor

A matrix has only one determinant, so either the given answer is wrong or the matrix you showed us is different from the one in your book's problem.

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