A matrix is a rectangular table used to organize numbers. For example, if you make a table of two classes' average scores in Korean, English, and math, that's a 2-row, 3-column matrix. The horizontal lines are called rows, and the vertical lines are called columns, and the number sitting in row i, column j is called the (i,j) entry of that matrix. If it has m rows and n columns, it's expressed as an "m×n matrix."
Adding and subtracting matrices isn't hard. Just take two matrices of the same size and calculate position by position. Scalar multiplication (kA), which multiplies an entire matrix by some number k, is just as simple — multiply every entry by k. So far, this is similar to the arithmetic we already know.
But multiplication is defined by a completely different rule. To multiply an m×n matrix A by a p×q matrix B, you first need A's number of columns (n) to equal B's number of rows (p). Only then can you calculate AB, and the result is an m×q matrix. The (i,j) entry of AB is found by multiplying row i of A with column j of B term by term and adding everything up.
The most remarkable part is that this multiplication gives a different result if you swap the order. The number multiplication we know always gives 15 whether it's 3×5 or 5×3, but with matrices, AB and BA are generally different from each other. Sometimes one is even defined while the other isn't defined at all. That's because a matrix isn't just a plain number — it represents something like a transformation or a relationship. "Apply A first, then apply B" and "apply B first, then apply A" naturally give different results when the order is different.
Matrices are also extremely useful in real life. For example, a table marking 1 if there's a direct route between three cities and 0 if there isn't is called an adjacency matrix, and multiplying this matrix by itself (M²) lets you calculate all at once "the number of ways to get somewhere by passing through exactly one intermediate stop." Follow relationships on social media, subway route maps, and even the link structure between web pages can all be represented and analyzed as matrices. The calculations behind rotating or scaling objects in computer graphics, and the calculations an AI uses to process huge amounts of data, ultimately speak the basic language of matrix operations too.