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How does a matrix product represent performing transformations in sequence?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, linear transformations are functions that map one vector to another while preserving vector addition and scalar multiplication. When performing computations, these transformations are represented by matrices.1

The product of two matrices represents the result of performing these transformations in sequence. Specifically, if you have two transformations represented by matrices $S$ and $T$ , the matrix product $S T$ corresponds to applying transformation $T$ first, followed by transformation $S$ (reading from right to left).1

Because matrix multiplication is generally not commutative—meaning $S T$ is rarely equal to $T S$ —the order in which you perform these transformations is critical. Changing the sequence of operations typically results in a different final transformation. This property is fundamental to applications such as computer graphics, where sequences of rotations and other geometric transformations are calculated using matrix products.21