scalar multiplication

Scalar multiplication means scaling a whole matrix up or down by one number. Pick a single number — the scalar — and multiply every entry by it, the way you might double a recipe by doubling every ingredient.

If k is a number and A is a matrix, then kA is the matrix whose entry in each position is k times the corresponding entry of A. The dimensions stay exactly the same; only the magnitudes change. The word “scalar” signals an ordinary number, in contrast to a matrix.

This operation is what lets you write things like 2A, -A (which flips every sign), or 0A (which is the all-zeros matrix). Combined with matrix addition, scalar multiplication gives you linear combinations like 3A - 2B, the basic moves of all linear algebra.