Cardinality of a set of matrices
Consider the set $S$ of $3\times3$ matrices with binary coefficients, that is, the coefficients are integers modulo $2$. Compute $|S|$.
I am not sure what is this question trying to ask. Am I right to say that it is asking in how many different unique way can we form a $3\times3$ matrix with coefficients of either $1$ or $0$?
If that is the case, is it $2^{3\times3}$?
Consider a set $ R = \{M \in S, M = M^T \}$ which is a subset of $S$ such that they are equal to their transposes. Compute $|R|$.
Am I right to say, given $M = M^T$, the coefficient $ij$, $M_{ij} = M_{ji}$?
$\endgroup$ 12 Answers
$\begingroup$Yes, you are correct so far.
The cardinality of $R$ is found thusly. Consider all matrices which are equal to their transpose, that is, symmetric matrices. Before you used the number of elements in the matrix to find the cardinality of the set, but now not all elements are independent. Indeed if we set the elements below the diagonal (3 of them) this will also set the elements above the diagonal due to symmetry. This leaves just the elements on the diagonal (3 of them). Thus the number of independent elements is 6 and, using your logic, the cardinality of $R$ is $2^6$.
$\endgroup$ 0 $\begingroup$Yes, both of your ideas are correct.
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