# How to prove the determinant of a matrix is zero?

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• April 13th 2010, 12:50 PM
Magnolia
How to prove the determinant of a matrix is zero?
The matrix M satisifies the following properties:
1. m_{ii}>0
2. m_{pq}m_{qp}=m_{pp}m_{qq}
3. m_{ij}m_{ji}=0 or m_{ii}m_{jj}, i,j not equal to p and q.
• April 14th 2010, 02:59 AM
HallsofIvy
Do a row reduction: look at what happens when you subtract, say, $\frac{a_{12}}{a_{11}}$ times the first row from the second. Try it with a 2 by 2 matrix first.
• April 14th 2010, 06:35 AM
Magnolia
Many thanks for your reply.

It holds when M is a 2 by 2 matrix.

I tried using row reduction, most items become same but there still exist some items uncertain.

In fact, I am perplexed by this problem for such a long time and I have tried using induction but failed.