Scalar Multiplication - Scaling Data

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Scalar Multiplication - Scaling Data Scale every entry of a price matrix at once by multiplying the matrix by a single number. A store keeps its product prices in a matrix A (each entry is a price in dollars). For the new season it raises every price by 20%, which means multiplying by the scalar c = 1.2 . Find the scaled price matrix cA , where and prices run from 25 up to 75 . Scalar multiplication touches every entry: each price a_ ij is replaced by , so the matrix keeps its 3×2 shape and only the values grow. Nice work! You scaled a whole price matrix with a single number. Here is what scalar multiplication does: Multiply every entry: — the scalar hits every entry, never just the diagonal. Here 1.2 turned 25 into 30 and 75 into 90 . Shape is preserved: cA stays the same 3×2 size as A ; only the values change. Distributes over a sum of matrices: c(A + B) = cA + cB . Distributes over a sum of scalars: (c + d)A = cA + dA . Edge scalars: (the zero matrix) and (no change). Scalar multiplication is a building block of linear algebra — combined with matrix addition it forms linear combinations, the engine behind transformations, eigenvalues, and scaling data sets like these prices.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.