Prove that If A,B are 3x3 tensors, then the matrix C=AB is also a tensor

[ReuvenD10]

Homework Statement

Prove that If A,B are 3x3 tensors, then the matrix C=AB is also tensor

Relevant Equations

the equations below in my solution

I try to solve but i have 1 step in the solution that I don't understand who to solve.

Below in the attach files you can see my solution, the step that I didn't make to prove Marked with a question mark.

thanks for your helps (:

 

[fresh_42]

It would help a lot if you would type in your question here (see how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/) instead of forcing people to download your pdf. An explanation what a $3\times 3$ tensor should be, if not a $3 \times 3$ matrix would be helpful, too.

To me it reads as:
Show that the product of two square matrices of equal size is again a square matrix of the same size.

 

[etotheipi]

To me it reads as:
Show that the product of two square matrices of equal size is again a square matrix of the same size.

It's funny, I interpreted it slightly differently. That $\mathcal{A}, \mathcal{B}, \mathcal{C}$ are some rank-2 and 3-dimensional tensors and we are asked to prove the tensor transformation properties, i.e. to show that if an equation in matrix representation $[\mathcal{C}]_{\beta_1} = [\mathcal{A}]_{\beta_1} [\mathcal{B}]_{\beta_1}$ holds with respect to basis $\beta_1$ then $[\mathcal{C}]_{\beta_2} = [\mathcal{A}]_{\beta_2} [\mathcal{B}]_{\beta_2}$ holds with respect to basis $\beta_2$. So for instance you have$$
\begin{align*}
\bar{c}_{\mu \nu} &= {\bar{a}_{\mu}}^{\gamma} \bar{b}_{\gamma \nu} = ({T^{\rho}}_{\mu} {T_{\sigma}}^{\gamma} {a_{\rho}}^{\sigma})({T^{\alpha}}_{\gamma}{T^{\beta}}_{\nu} b_{\alpha \beta}) \\

&= {T^{\rho}}_{\mu} {T^{\beta}}_{\nu} {a_{\rho}}^{\alpha} b_{\alpha \beta} \\

&= {T^{\rho}}_{\mu} {T^{\beta}}_{\nu} c_{\rho \beta}

\end{align*}
$$where the ${T^i}_j$ are the transformation coefficients from $\beta_1$ to $\beta_2$.

 

[fresh_42]

What is a matrix representation of a tensor? That doesn't make sense. A matrix is already a tensor. And any tensor other than $\sum u_k\otimes v_k$ isn't a matrix. $3$ by $3$ makes only sense for matrices. The rank should be completely irrelevant here.

 

[etotheipi]

Sure, yes I'm still doing some mental gymnastics to try and understand what is required. Generally a tensor doesn't have a 'matrix representation', but you can naturally map rank-2 tensors in $n$-dimensional space to $n \times n$ matrices, for ease of computation.

Anyway that's just how I interpreted it, I guess we need to wait for OP to explain what the question actually is asking.