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Fifa

Member

Registered: 2010-01-15

Posts: 6

Matrix algebra determinant proof help

The problem is... "If the entries in each row of an nxn matrix, A, add up to zero prove that the determinant of A is zero. (Hint: Consider the product AX where X is the nx1 matrix, each of whose entries is one.)"

Any help would be appreciated.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Matrix algebra determinant proof help

Hi Fifa,

I remember some results, so I think you can use something like this:

Hence you can add all the n columns and replace the first column, and you are done.
I have no idea how to use the hint in the question.

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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrix algebra determinant proof help

Hi Fifa;

I remember this one but I was not careful enough to write it down. This is what I would do.

Let x be a n x 1 column vector of ones.

The statement that all the rows add up to 0 can be rewritten as:

where 0 is a n x 1 column vector of 0's

If we invert A trying to solve the matrix equation and multiply both sides by A^(-1)

So the LHs is x

Now on the RHs A^(-1) . 0 = 0 So the system reduces to:

But from what was given.

This is a contradiction.

This means there is no solution to the linear set of equations. This only occurs when some rows of A are linearly dependent. This means A is not invertible therefore Det(A) = 0.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
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