In Brazil's curriculum students are taught matrices in high school. Here, however, there is no linear algebra or pre-calculus, therefore matrices end up being just tables with lots of "arbitrary" definitions.
Is there a way to motivate the definitions without teaching them things not on the curriculum?
Even without explicitly introducing the language of "linear maps", "vectors", and so on, you can still develop matrices as a shorthand for such maps, thought of as exchange rates.
Example:
Machine A can make 3 sprogs and 2 sprakets a day. Machine B can make 1 sprog and 3 sprakets a day. We summarize this data in a table of values:
$$\begin{bmatrix} 3 & 1 \\ 2 & 3 \end{bmatrix}$$
Where the first column represents machine A and the second column machine B, the first row sprogs, and the second row sprakets.
Now If my company buys $7$ machine As and $3$ machine Bs, what is my production capacity? This is a reasonable question for a 3rd grader, probably. All the matrix does is provide a structure for solving such a problem systematically, as:
$$\begin{bmatrix} 3 & 1 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 7 \\ 3 \end{bmatrix}$$
Multiplying a matrix by a vector should be defined to make the interpretation above valid.
Now say each sprog sells for $5$ and each spraket sells for $4$ dollars. Then we can get a new table telling us how much money machine A makes and how much money machine B makes. Again, this is a "third grade" problem, and we just introduce new notation for the problem:
$$\begin{bmatrix} 5 & 4\end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 3 \end{bmatrix}$$
Another example might be that each sprog lets another company build 2 cars and 3 buses, while a spraket lets them build 4 cars and 2 buses. So ultimately, to figure out how machine A and B relate to cars and buses, you can perform:
$$\begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}\begin{bmatrix} 3 & 1 \\ 2 & 3 \end{bmatrix}$$
At each stage you should be carefully tracking the meaning of these objects. The columns each represent one of a set of inputs, and each row represents an output. The entry at the $i^{th}$ column and $j^{th}$ row is the quantity of output $j$ produced by $1$ input $i$. Matrix multiplication corresponds to figuring out a new table if you are given two tables where the outputs of one are the inputs of the other.
That is to say, I suggest teaching the linear algebra in a particularly down to earth way, which is unlikely to intimidate students (or other teachers!) with words like "linear transformation" or "vector space".
Maybe you can try solving sets of linear equations (2 × 2 or 3 × 3) as a motivation example for matrices and their basic manipulations.
Short of changing the curriculum :-), maybe you can use magic squares, or even Sudoku, to help motivate. See, e.g., "From Magic Squares to Sudoku". Here is Dürer's $4 \times 4$ magic square:
One of the favorite mathematical activities of many students is "cancellation." An example is when you are asked to reduce a complicated fraction, and you can do so by "factoring out" a large number from both numerator and denominator, leaving "lowest terms."
If you can convince students that matrices are really representative of systems of equations, they can have a lot of fun reducing the systems of equations to lowest terms. Using "low echelon row reduction" for instance, you can create a diagonal matrix, where one equation is a number of x, the equation above it relates to only x's and y's, etc.
Another thing to do is to show how matrices determine polyhedrals in space. And how the related determinants "reduce" to the volumes of these polyhedrals.
Surely you teach complex numbers in the curriculum.
One thing to do with matrices is to use them to create other mathematical objects, or at least a model of the object. In particular, we can use $2 \times 2$ real matrices of the form $\left[ \begin{array}{cc} x & -y \\ y & x \end{array}\right]$ to model the complex number $x+iy$. Matrix addition and multiplication then correspond naturally to the addition and multiplication of complex numbers and the complex number $1$ corresponds naturally to the matrix $\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right]$ etc. In addition, in this discussion you can either look at the formula $\frac{1}{z} = \frac{\bar{z}}{z\bar{z}}$ as a method to find the multiplicative inverse of a complex number, or you can use the $2\times 2$ inverse formula (if that is in the scope of the discussion).
Probably Steven Gubkin's answer is more in-line with your goals. But, I think highschool is the right place to start talking about this circle of ideas.
Try to handle and solve for the $x$'s conceptually this system,
or this one
$$A\mathbf x = \mathbf b$$
where $A$ is a $K\times K$ matrix, and $\mathbf x, \mathbf b $ are $K \times 1$ vectors, $K=10^6$. Hmm, why not set $K$ equal to one trillion? Or a hundred thousand trillion.
And when the Matrix Algebra interrelations start to become more and more intricate and impressive, beyond the basic level of solving a linear system of equations, and still the dimensions can be as large as we want, this is how I fell for Linear Algebra and matrices: once I realized that this dance remains elegant irrespective of the size of the entities we dance with (while also being able to visualize somehow the enormity of them -in most other mathematics, the object dimensions are a bit too abstract for most people).
It's like Circus Human Acrobatics perfectly performed by elephants. Or galaxies.
