I am taking an into to math in data science class. The issue is that it's been 15 years since my last math class and my abstract evaluation skills are rusty. The problem is as follows:
Consider the set $\mathscr{G}$ of 3×3 matrices defined as follows:
$$ \mathscr{G} = \Bigg\{\Bigg[\begin{matrix}1&x&z\\
0&1&y\\
0&0&1\end{matrix} \Bigg] \in \mathbb{R}^{3×3} \Bigg| x,y,z \in \mathbb{R} \Bigg\}$$
We define $\cdot$ as the standard matrix multiplication.
Is $(\mathscr{G}, \cdot)$ a group? If so, is it Abelian?
I guess I just mainly don't understand what the question is asking at the end. The matrix by itself cannot be a group. So is the second part saying "This given matrix, with another 3×3 matrix that is in the realm of real numbers?"
If so, when looking at the criteria for Abelian groups: Binary operation on G, Associativity, Identity Element, Inverses, and Commutativity:
I know that multiplying this 3×3 matrix by another 3×3 matrix will give the proper shape, but how do I demonstrate a binary operation if I can't know that x,y or z will be 0?
I know that matrix multiplication is associative.
Then I get very confused because the Identity element for a 3×3 matrix would mean x.y,z are all equal to zero, right?
At this point, my brain is hurting and I can't understand how to get the inverse of the matrix. A 2×2 is simple, but I keep getting different answers every time I try with this 3×3 matrix. I think this is just a simple math error, but I am not sure.
So then with commutativity, I don't ever find a solution where a 3×3 matrix is commutative with the format of the given matrix.
So am I correct that at least the group isn't Ablian? How do I find out if it's a group?
Best Answer
$\mathcal G$ is a set of matrices, and not a single matrix.
Does this set of matrices, together with with the operation of matrix multiplication form a group?
Is it closed? Is it associative? Is there an identity matrix? Does every matrix in $\mathcal G$ have an inverse in $\mathcal G$?
Associativity and the identity matrix are inherited with the operation, so we don't really need to test those. Matrix mulitplication is generally not commutative, but in special cases, it is.
For closure, pick two generic matrices in the set, multiply them together and what do you get?
$\begin{bmatrix} 1&x_1&y_1\\0&1&z_1\\0&0&1\end{bmatrix}\begin{bmatrix} 1&x_2&y_2\\0&1&z_2\\0&0&1\end{bmatrix} = \begin{bmatrix} 1&x_1 + x_2&y_1 + y_2 + x_1z_2\\0&1&z_1 +z_2\\0&0&1\end{bmatrix}$
Is it in $\mathcal G$? If we can find an inverse, is that in $\mathcal G$?
Does it look like this is one of those cases where matrix multiplication commutes?