Is it Algebra at all?

Geometry Level 5

Find the number of different 3 × 3 3 \times 3 matrices with determinant value 0, such that each of its entries is either + 1 +1 or 1 -1 , but there are no two rows that are identical.


The answer is 144.

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1 solution

Soumava Pal
Mar 4, 2016

This problem is really an interesting example to show that we need not do algebra problems by algebra alone, or geometry problems by geometry alone, and that mathematics as a whole is a subject and there can really be no sharp boundaries between the disciplines. I love how this problem can be posed in an algebraic way, interpreted with the help of vectors, visualized with the help of 3-D geometry, and answered with the help of combinatorics.

We note that any such matrix, is the scalar triple product of vectors, a a , b b , c c where the x,y,z-components of the three vectors are +1 or -1.

But the scalar triple product of three vectors is the volume of the parallelopiped with three adjacent sides along the three vectors, and if the volume is 0, that means that the 3 vectors are coplanar.

Any point whose position vector has x,y,z-components +1 or -1 is a vertex of the cube of side length 2 units,centred at the origin and sides parallel to the axes.

So we are looking for 3 coplanar vectors among the 8 such vectors possible. Taking the origin as O, we observe that any triple of 3 coplanar vectors from the origin to the vertices will lie on the plane (containing 4 vectors) determined by two opposite face diagonals and the corresponding sides as shown below.

Now such a plane can be chosen in 6 distinct ways, 2 for each pair of opposite faces, and from the 4 vectors on each plane, 3 vectors can be chosen in 4 C 3 = 4 4C3=4 ways, so the total number of ways of choosing three such vectors is 4 6 = 24 4*6=24 ways.

But in the matrix the 3 vectors can be arranged from top to bottom in 3 ! = 6 3!=6 different ways, so the answer is 24 6 = 144 24*6=144 .

I'm a bit confused by your count: Are you including matrices with two or three identical rows?

Otto Bretscher - 5 years, 3 months ago

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@Otto Bretscher

I am so sorry I forgot to mention that.

I have edited the problem. Hope it is okay now.

Soumava Pal - 5 years, 3 months ago

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There is still a small problem: You might have distinct rows but two identical columns. Better to say: "Matrices with two or more identical rows are not to be counted."

I liked the original problem, without excluding identical rows. Will you post that one too? Or shall I?

Otto Bretscher - 5 years, 3 months ago

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@Otto Bretscher @Otto Bretscher

Oh, wow, that's nice. Yeah, comrade, it would be great if you could post it, please do so, because I have my physics exam in 5 hours at school. So I can't post it right now.

:)

Soumava Pal - 5 years, 3 months ago

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@Soumava Pal Good luck with the exam! You can do it, Comrade! (and don't forget to omit "(or columns)" to make the problem fully correct.)

Otto Bretscher - 5 years, 3 months ago

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@Otto Bretscher Thank You.

Yes, I was just doing it.... :P when the comment came.

Soumava Pal - 5 years, 3 months ago

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