Is this one hard?! Part 2

Algebra Level 3

Let M M be a set of real n × n n \times n matrices such that

  • I ∈ M I \in M , where I I is the n × n n \times n identity matrix;

  • if A ∈ M A \in M and B ∈ M B \in M ,then either A B ∈ M AB \in M or − A B ∈ M -AB \in M , but not both;

  • if A ∈ M A \in M and B ∈ M B \in M ,then either A B = B A AB=BA or A B = − B A AB=-BA ;

  • if A ∈ M A \in M and A ≠ I A \ne I ,there is at least one B ∈ M B \in M such that A B = − B A AB=-BA .

Is M M contains at most n 2 n^2 matrices?

Yes No

Gia Hoàng Phạm by Gia Hoàng Phạm , 19, Vietnam

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1 solution

The orthogonal group of order n has two components. One which includes the identity transformation (sometimes called the special rotation group) and the other component includes a reflection. When represented by real n × n n\times n matrices, the first group is has a determinant of 1 and the other component has a determinant of -1 (the reflection). The size of the first component is n ( n + 1 ) 2 \frac{n(n+1)}{2} and of the other component n ( n − 1 ) 2 ] \frac{n(n-1)}{2}] . The sum of these two triangular numbers is n 2 n^2 .

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