Multiplication of a Matrix by a Scalar Value

Multiplying a matrix by a scalar (an ordinary number) is the simplest matrix operation: every single element of the matrix gets multiplied by that number. If $k$ is a scalar and $A = [a_{ij}]$ is a matrix of order $m \times n$, then $kA$ is another matrix of the same order $m \times n$, obtained by scaling every entry:

Imagine a baker's ingredient list written as a matrix, with one row per bread type and one column per ingredient. If the baker decides to triple every recipe for a busy Sunday, the new ingredient matrix is simply $3A$ — every quantity scaled by $3$, with the shape of the matrix completely unchanged. Scalar multiplication has some natural properties: for matrices $A, B$ of the same order and scalars $k, p$,

$k(A + B) = kA + kB, \qquad (k+p)A = kA + pA, \qquad k(A - B) = kA - kB$ …