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 $k$ ?

#Combinatorics #MathProblem #Math

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 :)

Battista Lonardi - 7 years, 11 months ago

yup! Thanks! :)

Advitiya Brijesh - 7 years, 11 months ago

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

Bob Krueger - 7 years, 11 months ago

How would you define a symmetric matrix?

Tim Vermeulen - 7 years, 11 months ago

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

Advitiya Brijesh - 7 years, 11 months ago

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.

Bob Krueger - 7 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$