# Permutations in Matrix!

Question-Let $$A$$ be the set of all $$3$$x$$3$$ symmetric matrices whose entries are $$1, 1, 1, 0, 0, 0, -1, -1, -1$$. $$B$$ is one of the matrix in set $$A$$.

Number of such matrices $$B$$ in set $$A$$ is $$k$$. Then, what is the value of $$k$$?

Note by Advitiya Brijesh
5 years ago

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## Comments

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A $$3 \times 3$$ symmetric matrix follows this pattern: $\begin{pmatrix} a & d & e \\ d & b & f \\ e & f & c \end{pmatrix}$ So we have to choose which numbers are $$a$$, $$b$$, $$c$$, $$d$$, $$e$$ and $$f$$. Firstly, we'll focus on the principal diagonal: $\begin{pmatrix} a & & \\ & b & \\ & & c \end{pmatrix}$ If we had $$a = b \neq c$$, the third entry of $$a$$ wouldn't have a symmetric entry. For the same reason, we can't have $$a \neq b = c$$ nor $$a = c \neq b$$. If we had $$a = b = c$$, then we couldn't build a symmetric matrix: we could choose a number for $$d$$ and $$e$$, but there couldn't be symmetry in the remaning places. Hence $$a$$, $$b$$ and $$c$$ are all different numbers, and we have $$3!$$ possibilities. Now, we'll focus on the other numbers, outside the principal diagonal: $\begin{pmatrix} & d & e \\ d & & f \\ e & f & \end{pmatrix}$ Since $$a$$, $$b$$ and $$c$$ are all different numbers, we have 2 entries of $$-1$$, $$0$$ and $$1$$ left. It's obvious that $$d$$, $$e$$ and $$f$$ must also be all different numbers, and we have $$3!$$ possibilities for them too. So the cardinality of the set $$A$$ is $${ \left( 3! \right) }^2 = 36$$. I'm quite sure this is the right answer, but I'm open to corrections :)

- 5 years ago

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yup! Thanks! :)

- 5 years ago

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Here's a better phrasing of the question: How many symmetric 3x3 matrices exist, with entries consisting of three 1's, three 0's, and three -1's.

Also, I'm going to put this answer in code in case any users don't want to see it: (5^2 - 13)*3

- 5 years ago

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How would you define a symmetric matrix?

- 5 years ago

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Think about the diagonal that goes from the upper left to the bottom right corners. A symmetric matrix has symmetry about this line. This really only works for square matrices.

- 5 years ago

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If $$A$$ is a matrix of some order. Then, $$A$$=$$A^T$$. Where, $$A^T$$ denotes the transpose of matrix $$A$$. For Transpose of matrix, see here, http://en.wikipedia.org/wiki/Transpose

- 5 years ago

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