Calculating Determinants: A Guide to Finding the Correct Answer

[60051]

Homework Statement

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?

 

[berkeman]

Homework Statement

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?

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

.

 

[Mark44]

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.

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

It should be
1 [19 - 7] - (-3) [4 - 0]
= 12 + 12
= 24

 

[berkeman]

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

 

[Mark44]

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

 

[60051]

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

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

 

[Mark44]

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.